STUDY OF CHORDAL GRAPHS
ABSTRACT THESIS
SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF
Bottor of $t)t(ogapI)p IN APPLIED MATHEMATICS
BY PARVEZ ALI
Under the Supervision of Dr. Merajuddin
DEPARTMENT OF APPLIED MATHEMATICS Z.H. COLLEGE OF ENGINEERING & TECHNOLOGY ALIGARH MUSLIM UNIVERSITY ALIGARH (INDIA) 2008 Siiidv ofChordal Graphs Absract
Abstract
Perfect graphs are an active area of current research for their potenticJ application to many real life problems and for their nice combinatorial structures as well. The notion of perfect graph was introduced by Berge [3] in 1960. In search of certificates for perfection, Berge [2] made two conjectures concerning perfect graphs. The first one was proved by, Lovasz in 1972 and since then called the Perfect Graph Theorem. The second conjecture known as the Strong Perfect Graph Conjecture (SPGC) and attempts to prove it, contributed much to the development of graph theory in the past forty years. Chudnovsky et al. [5] recently were able to prove the SPGC in its fiill generality. After remaining unsolved for more that forty years it is now called the Strong Perfect Graph Theorem (SPGT). A second motivation for studying perfect graphs besides the SPGT is their nice algorithmic properties. While the problems of finding the, (i) clique number, (ii) chromatic number, (iii) stability number and (iv) clique covering number of a graph are NP-hard in general [9], they can be solved in polynomial time for perfect graphs. It is still an open problem to find a polynomial time algorithm to color perfect graphs or to compute the clique number, stability number, clique covering number of a perfect graph. Berge [4] showed that many familiar classes of graphs such as chordal graphs, comparability graphs, interval graphs, unimodular graphs and line graphs of bipartite graphs are perfect. Foldes and Hammer established that split graphs belong to the class of chordal graphs. Thus these graphs are also perfect. Various others important classes of perfect graphs are weakly chordal graphs, quasi-chordal graphs, strongly chordal graphs and perfectly orderable graphs. In fact the world of perfect graphs has grown to include over 200 special graph classes [1]. Sliidv ofChordal Graphs Absract
The results of Comeil [7] motivate the study of various classes of self- complementary perfect graphs and self-complementary imperfect graphs. In this thesis we study self-complementary perfect graphs and some of its subclasses namely, self-complementary chordal graphs, self-complementary weakly chordal graphs, self-complementary quasi-chordal graphs, self- complementary perfectly orderable graphs, self-complementary brittle graphs and P5-free self-complementary weakly chordal graphs. The class of self-complementary perfect graphs enjoys both the properties of self-complementary graphs and the perfect graphs. For example, one of the nice property of self-complementary perfect graph is that all the four classical graph parameters i.e. clique number, chromatic number, stability number and clique covering number becomes equal to each other. The class of self-complementary graphs, chordal graphs, quasi-chordal graphs, weakly chordal graphs, perfectly orderable graphs and brittle graphs had been studied in the literature [1]. Chordal graphs were introduced by Hajnal and Suranyi [12]. They proved that these graphs are a-perfect. Berge [3] proved that j-perfectness of these graphs. These graphs find applications in evolutionary trees, scheduling, solutions of sparse systems of linear equations and computational biology. Many graphs problems including the four classical optimization problems that are NP-hard in general graphs can be solved in polynomial time in chordal graphs [9]. A concept called Perfect Elimination Ordering (PEO) is important in chordal graphs. It turns out that all the existing chordal graph recognition algorithms and many optimization problems in chordal graphs make use of PEO [10].
Quasi-chordal graphs were introduced by Voloshin in [15] and weakly chordal graphs were studied by Hayward [13] as a natural generalization of chordal graphs. We study this class in detail in the thesis. Sliidv ofChordal Graphs Absract
Chvatal [6] defined the class of perfectly orderable graphs, motivated by the graph coloring problem. This class of perfect graphs contains chordal graphs, quasi-chordal graphs and various others [1]. For a recent survey, see the chapter -7 by Hoang in [14]. The class of brittle graphs is one of the first subclass of perfectly orderable graphs.
Spectral graph theory is the study of the spectra of certain matrices defined by a given graph. The most common matrix investigated has been the adjacency matrix. Some of the common problem studied in the graph spectra are study of cospectral graphs, bounds on eigenvalues, energy of graphs and various others [8]. The concept of graph energy was introduced by Gutman [11], in connection to the so called total n- electron energy. There are numerous upper and lower bounds for the graph energy as reported in the literature [11].
In this thesis we study some structural and algorithmic aspects of certain subclasses of sc perfect graphs. In chapter 1 we present some definitions, introduce perfect graphs and its various subclasses and give a brief introduction of the outline of the thesis. In chapter 2, we study sc chordal and sc weakly chordal graphs. We study the relation between the two-pair and sc weakly chordal graphs and report a lower and upper bound for this. Using this bound we propose an algorithm for finding chordless cycle d, in sc weakly chordal graph which is not chordal. We also study the recognition problem of sc chordal and sc weakly chordal graphs and propose algorithms in both the cases. Using these algorithms we compile the catalogue of sc chordal and sc weakly chordal graphs up to 17 vertices. In chapter 3, we extend our study to the other subclasses of sc perfect graphs, such as sc brittle graphs, sc quasi-chordal graphs and Ps-free sc weakly chordal graphs. One of the important relation between all these 3 classes is that they are subclasses of sc perfectly orderable graphs. Sltidx of Chorda! Graphs Absract
The problem of finding a maximum clique, a minimum colouring, a maximum stable set and a minimum clique cover of a graph is known as optimization problems. These four problems are NP- hard in general [9]. In chapter 4, we study these optimization problems for sc perfect graphs and their subclasses namely, sc chordal graphs, sc weakly chordal graphs, sc quasi- chordal, sc brittle graphs. In general obtaining the exact value of the chromatic number of a graph is quite difficult. However we give an upper and lower bounds for the chromatic number of sc perfect graph. We have studied spectral properties of sc perfect graphs and its subclasses in chapter 5. We obtain a result for interlacing Theorem when the graph is sc. We show that the class of sc and sc weakly chordal graphs cannot be determined by its spectrum. We prove that the least positive integer for which there exist non-isomorphic cospectral sc weakly chordal graph is 12. We report many results related to spectrum of sc perfect graphs and its subclasses.
References 1. A.Barndstadt, V.B. Le and J.P. Spinrad, Graph Classes: A Swvey, SI AM Philadelphia (1999). 2. C.Berge, Fdrbung von Graphen, deren sdmtliche hzw. deren imgerade Kreiese stairsin, Wiss. Z. Martin-luther Univ., Halle-Wittenberg Math- Natur., Reihe(1961) 114-115 3. C.Berge, Les problemes de colorations en theorie des graphes, Publ. Inst. Stat. Univ. Paris, 9(1960) 123-160. 4. C.Berge, Some classes of perfect graphs, Graph theory and Theoretical physics. Academic Press, New York, (1967) 155-165. 5. M.Chudnovsky, N.Robertson, P.D.Seymour and R.Thomas, The Strong Perfect Graph Theorem, Annals of Mathematics, 164 (2006) 51-229. 6. V.Chvatal, Perfectly ordered graphs, Ann. Disc. Math.,21 (1984) 63-65. 7. D.G.Comeil, Families of graphs complete for the Strong-Perfect Graph Study ofChordal Graphs Absract
Conjecture, J. Comb. Theory Series B, 10(1986) 33-40. 8. D.Cvetcovic, M.Doob and H.Sachs, Spectra of Graphs-Theory and Application, Academic Press, New York, (1980). 9. R.Garey and D.S.Johnson, Computers and Intractability: A Guide to the Theory ofNP-Completeness, (W.H.Freeman, New York, 1979). 10. M.C.Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, (1980). 11. I.Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungszenturm Graz, 103(1978) 1-22. 12. A.Hajnal and J.Suranyi, Uber die Aflasung von Graphen in vollstandige Teilgraphen, Ann. Univ. Sci. Budapest Etovos. Sect. Math., 1(1958) 113-121. 13. R.B.Hayward, Weakly triangulated graphs, J. Comb. Theory Series B, 39 (1985) 200-209. 14. J.L.Ramirez and B.A.Reed, Perfect graphs, John Wiley and Sons, (2000). 15. V.I.VoIoshin, Quasi-triangulated graphs, Preprint, 5569-81, Kishinev state university, Kishinev, Moldova, 1981( in Russian). T6506 STUDY OF CHORDAL GRAPHS
THESIS
SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF
Bottor of ^t)iIo£iopI)P IN APPLIED MATHEMATICS
BY PARVEZ ALI
Under the Supervision of Dr. Merajuddin
DEPARTMENT OF APPLIED MATHEMATICS 2.H. COLLEGE OF ENGINEERING & TECHNOLOGY ALIGARH MUSLIM UNIVERSITY ALIGARH (INDIA) 2008
Department of Applied Mathematics Faculty of Engineering (Z. H. College of Engg. & Technology) ALIGARH MUSLIM UNIVERSITY ALIGARH—202 002 (U P.\ INDIA EPBAX: 700920/21/22/37 Internal Phone : 453. Office : 452
Dated.
Certificate
Certified that the thesis entitled "Study of Chordal Graphs" being submitted by Mr. Parvez Ali, in partial fulfilment of the requirement for the award of the degree of Doctor of Philosophy, is a record of his own work carried out by him under my supervision and guidance. The matter embodied in this thesis has not been submitted for the award of any other degree or diploma.
(DR. MERAJUDDIN)
Supervisor Acknowledgements
'First and foremost, I render infinite tdan^s to Qod, Wfio sustains
me and gave me the strength, determination and discipCine to compCete
the thesis.
'Dr. Merajuddin has Been more than a thesis supervisor to me. I
express my deep sense of gratitude to him for his constant encouragement
and exfeCCent supervision. His vaCuaSfe suggestions and active patience
throughout the thesis wor^ have heCped in improving the presentation of
the thesis. I am very than^uC to him for SuiCding in me a right ^nd of attitude towards research. "Kis warmth, friendliness, and fatherly affection have undoubtedly made my association with him a cherished phase in my life.
The present dissertation is nothing But an encapsulation of my parents Blessing who have Been a source of constant inspiration to me at every stage during my research wor^ without their love and Blessing this wor^ could not Be completed. To my family, thanks so much for all of your love, hjndness, encouragement and support.
I have had a good fortune to wor^with my colleagues and friends
(Dr.Madhu^ar Sfiarma, Syed JLjaz %areem %irmani, Vmutul Samee and MoUcfJizeem. It was a pCeasure to wor^witfi them and I tHan^tHem aCC for having shared my experience and thoughts throughout the fast severaC years.
Infact, merefy mentioning names cannot pay thanks 6ut I am grateful to the entire members of JLppCied Mathematics ^Department,
AMU, Migarh.
I am aCso than^uC to I.I.'T. %anpur LiSrary staff for their co operation in complementing this tash^
Last 6ut certainly not least, I would li^e to than^ the _^ligarh
Muslim Vniversity, Jtligarh for providing me all the facilities.
$n-
{PARVEZ ALI) Contents
1 Introduction 1-22 1.1 General introduction 1 1.2 Outline of the thesis 14 1.3 Basic definitions and notations 16 2 On sc chordal and sc weakly chordal graphs 23-54
2.1 Introduction 23 2.2 Basic results 23 2.3 sc weakly chordal graphs and two-pairs 24 2.4 Recognition ofsc chordal and sc weakly chordal graphs 31 2.5 Catalogue compilation of sc chordal and sc weakly chordal graphs 40 2.6 Construction ofsc chordal and sc weakly chordal graphs 44
3 On the recognition of some classes of sc perfectly orderable graphs 55-82 3.1 Introduction 55 3.2 Sc brittle graphs and its recognition 55 3.2.1 Catalogue compilation ofsc brittle graphs 63 3.3 Quasi-chordal and sc quasi-chordal graphs 64 3.3.1 Recognition of sc quasi-chordal graphs 67
3.3.2 Catalogue compilation ofsc quasi-chordal graphs 75 3.4 Ps-free weakly chordal and Ps-free sc weakly chordal graphs ... 75
3.4.1 Recognition ofP^-free sc weakly chordal graphs 78
3.4.2 Catalogue compilation ofPyfree sc weakly chordal graphs 81 4 On the optimization of some classes of sc perfect graphs 83-107 4.1 Introduction 83 4.2 On the chromatic number ofscperfect graphs 84 4.3 Contraction method and optimization algorithms 86 4.3.1 Optimizing Algorithms for sc weakly chordal graphs 87 4.4 Optimization Algorithms for the class ofsc perfectly orderable 102 graphs 4.4.1 Optimizing Algorithm for sc quasi-chordal graph andsc brittle 103 graphs 4.4.2 Algorithm for finding the solution of optimization problems for sc quasi-chordal graphs and sc brittle graphs without given perfect order 107
5 On the spectrum of some classes ofsc perfect graphs 108-131 5.1 Introduction 108 5.2 Sc perfect graph and Interlacing Theorem 109 5.3 Non-DS sc and sc weakly chordal graphs 112 5.4 On the cospectrality ofsc chordal and sc weakly chordal 114 graphs 5.5 Equi-energetic sc chordal and sc weakly chordal graphs 117 5.6 Maximal energy of sc perfect graphs 120 5.7 Hyper-energetic sc chordal and sc weakly chordal graphs 125
References 132-144 List of Figures
Figure-1 4 Figure-2.1 24 Figure-2.2 27-28 Figure-2.3 (a, b) 34 Figure-2.4 (a, b) 39 Figure-2.5 (a, b) 41 Figure-2.6 (a-g) 42 Figure-2.7 (a~g) 43 Figure-2.8 (a ~g) 49-50 Figure-2.9 (a~g) 54 Figure-3.1 (a, b) 61 Figure-3.2 (a-h) 62 Figure-3.3 65 Figure-3.4 66 Figure-3.5 66 Figure-3.6 66 Figure-3.7 66 Figure-3.8 (a, b) 73 Figure-3.9 (a~h) 74 Figure-3.10 76 Figure-3.11 (a. b) 80 Figure-4.1 90 Figure-4.2 (a-f) 91 Figure-4.3 (a, b) 93
Figure-4.5 (a, b) 95
111 Figure-4.6 96 Figure-4.7 (a, b, c) 97 Figure-4.8 100 Figure-4.9 (a-g) 101 Figure-4.10 104 Figure-4.11 (a, b) 106 Figure-5.1 (a, b) 113 Figure-5.2 114 Figure-5.3 (a, b) 116 Figure-5.4(a-p) 128-129 Figure-5.5(a-p) 130-131 Figure-5.6 124 Figure-5.7 126
IV List of Algorithms
Algorithm 2.1 29 Algorithm 2.2 30 Algorithm 2.3 33 Algorithm 2.4 37 Algorithm 2.5 45 Algorithm 2.6 52 Algorithms.! 58 Algorithm 3.2 59 Algorithm 3.3 70 A Igorithm 3.4 71 Algorithm 3.5 79 Algorithm 4.1 89 Algorithm 4.2 99 Algorithm 4.3 105 CHAPTER ONE Chapter 1 Introduction
Chapter-1
Introduction
1.1 General introduction
Perfect graphs are an active area of current research for their potential application to many real life problems and for their nice combinatorial structures as well. The notion of perfect graph was introduced by Berge [7] in
1960. In search of certificates for perfection, Berge [6] made two conjectures concerning perfect graphs. The first one was proved by, Lovasz [90] in 1972 and since then called the Perfect Graph Theorem. It states that a graph is perfect if and only if its complement is perfect. A slightly stronger version of this theorem, the Semi-Strong Perfect Graph Theorem was proved by Reed
[112] in 1987.
Berge's second conjecture stated that a graph is perfect if and only if it contains, as an induced subgraph, neither an odd cycle of length at least five nor its complement. This conjecture became known as the Strong Perfect
Graph Conjecture (SPGC) and attempts to prove it contributed much to the development of graph theory in the past forty years. SPGC has led to the definitions and study of many new classes [8] of graphs for which the correctness of this conjecture has been verified. For several of these classes the
SPGC has been proved by showing that every graph in this class can be obtained from certain simple perfect graphs by repeated application of Chapter I Introduclion
perfection preserving operations. By using this approach Chudnovsky,
Robertson, Seymour and Thomas [21] were recently able to prove the SPGC in
its full generality. After remaining unsolved for more that forty years it is now
called the Strong Perfect Graph Theorem (SPOT).
A second motivation for studying perfect graphs besides the SPOT are their nice algorithmic properties. While the problems of finding the, (i) clique number, (ii) chromatic number, (iii) stability number and (iv) clique covering number of a graph are NP-hard in general [48], they can be solved in polynomial time for perfect graphs. This result is due to Grotschel, Lovasz and
Schrijver [59]. Unfortunately, their algorithms are based on the ellipsoid method and are therefore mostly of theoretical interest. It is still an open problem to find a polynomial time algorithm to color perfect graphs or to compute the clique number, stability number, clique covering number of a perfect graph.
Graph classes are particular sets of graphs, based on a structural characterization, or a forbidden induced subgraph. The study of graph classes sometimes allows us to solve problems on these restricted classes that cannot be solved on graphs in general. Due to this, many researchers, in order to prove
SPGC, started to study the subclasses of perfect graphs. Berge [8] showed that many familiar classes of graphs such as chordal graphs, comparability graphs, interval graphs, unimodular graphs and line graphs of bipartite graphs are perfect. Foldes and Hammer [45] established that split graphs belong to the Chapter I Inlrodiiclion
class of chordal graphs. Thus these graphs are also perfect. The idea of
splittance of a graph is the minimum number of edges to be added or deleted in
order to produce a split graph, was introduced by Hammer and Simmone [63].
According to this definition split graphs are those graphs, whose splittance is
zero. Various other important classes of perfect graphs are weakly chordal
graphs, quasi-chordal graphs, strongly chordal graphs and perfectly orderable
graphs. In fact the world of perfect graphs has grown to include over 200
special graph classes [3].
A class of graphs is said to be valid for a conjecture if the conjecture is
true for this restricted class of graphs. Many special classes of graphs called
/C| rfree graphs [101], toroidal graphs [58], (/C4-e)-free graphs [102], planar
graphs [143] and many other classes of graphs [19], [40], [88] and [142] were
shown to be valid classes of SPGT. A class of graphs is said to be complete for
a conjecture if the truth of the conjecture on this class implies the truth of the
conjecture in general.
The results of Comeil [33] motivate the study of various classes of self-
complementary perfect graphs and self-complementary imperfect graphs. In
this thesis we study self-complementary perfect graphs and some of its
subclasses namely, self-complementary chordal graphs, self-complementary
weakly chordal graphs, self-complementary quasi-chordal graphs, self- complementary perfectly orderable graphs, self-complementary brittle graphs Chapter 1 Introduction
and Ps-free self-complementary weakly chordal graphs. The relationships
between all these classes are shown in figure-1.
Perfect graphs Perfectly orderable graphs
/'5-free weakly chordal graphs
Weakly choj sraohs
Quasi-chordal graphs Chordal graphs Self-complementary Figure-1 graphs
The class of self-complementary perfect graphs enjoys both the properties of self-complementary graphs and perfect graphs. For example, one of the nice property of self-complementary perfect graphs is that all the four classical graph parameters i.e. clique number, chromatic number, stability number and clique covering number become equal to each other. The class of self-complementary graphs, chordal graphs, quasi-chordal graphs, weakly chordal graphs, perfectly orderable graphs and brittle graphs had been studied
...a.S^^' Chapter I Introduction
in the literature [3]. We discuss briefly about the various results available in
these classes.
Before that, we note the following remark:
In literature chorda! graphs appear under various names (triangulated
graphs, rigid circuit graphs, monotone transitive graphs and perfect
elimination graphs) as early as the sixties. Later many researchers called it as
triangulated graphs. However, recently people have begun to largely prefer the term chordal graphs. Perhaps 'triangulated' can be confused with another similar term 'triangulation'. Similarly weakly chordal graphs were originally called as weakly triangulated graphs, but later perhaps due to same reason it is now widely known as weakly chordal graphs. Now for the sake of uniformity in the thesis we use quasi-chordal graphs instead of quasi-triangulated graphs as known in the literature. Moreover, sc refers to self-complementary in all the contexts throughout the thesis.
Self-complementary graphs (sc graphs)
The existence problem of sc graphs was solved independently by Ringel
[113] and Sachs [119]. They proved that sc graphs with n vertices exists if and only if/? = Ap or n = 4p+l for some positive integer/?. Ringel [113] obtained algorithms to construct sc graphs with 4/? and Ap+] vertices. Algorithms for constructing sc graphs with 4/; and Ap+\ vertices were also obtained by Gibbs
[50] by modifying Ringel's algorithms. Harary [65] posed the problem of counting non-isomorphic sc graphs given the number of vertices. A complete Chapter I Inlroduction
solution for this problem was given by Read [109] and [110]. His method is
based on enumeration theory originated by Redfieid [111] and Poiya [103],
developed further by DeBruijn [38], [39] and Haraty et al. [64]. The number of
sc graphs (non-isomorphic) S,, for a given number of vertices n is given in the
following table-1 for n < 17. An asymptotic formula for Sn as «-* oo was
derived by Palmer [99]; see also Robinson [115], Sridharan [135] and Schwenk
[122].
n I 4 5 8 9 12 13 16 17
s„ I I 2 10 36 720 5600 703760 11220000
TabIe-1
The problem of deciding whether two given graphs are isomorphic or
not is called the isomorphism problem. Isomorphism of sc graphs and regular sc graphs was discussed by Colboum et al. [29] and [30]. They proved that the isomorphism of these classes is polynomial equivalent to the general graph isomorphism. The problem of deciding whether a given graph is sc or not is called the recognition problem of sc graphs. Colboum et al. [29] proved that the recognition of sc graphs is polynomial equivalent to general graph isomorphism. Catalogue of sc graphs with small number of vertices was compiled by Alter [1], Faradzhev [43], Kropar et al. [87], Morris [95], and
Venkatachalam [145]. Clapham and Kleitman [27] had obtained a necessary and sufficient condition for a degree sequence to be the degree sequence of a sc Chapter I huroduclion
graph. Further results on the degree sequences of sc graphs are given in [20]
and [28]. For various other results on sc graphs we refer to Farrugia [44], "Self-
complementary graphs and generalization: A comprehensive reference manual"
which contains 410 references.
Chordal and Quasi-chordal graphs
The class of chordal graphs is classical and extensively studied in
literature [5], [18], [41], [49], [50], [54], [55], [62], [81], [114], [117], [118],
[125], [130] and [141]. Chordal graphs were introduced by Hajnal and Suranyi
[62]. They proved that these graphs are o-perfect. Berge [7] proved that x- perfectness of these graphs. These graphs find applications in evolutionary trees [18], scheduling [100] and solutions of sparse systems of linear equations
[117]. Recently, in order to represent the model of protein interaction network and multidomain proteins, chordal graphs become an important ingredient in the field of computational biology [104]. Many graphs problems including the four classical optimization problems that are NP-hard in general graphs can be solved in polynomial time in chordal graphs [48] though testing Hamiltonicity
[31], determining the domination number [15] and other problems [48] and
[83] are NP-complete for this class too. A concept called Perfect Elimination
Ordering (PEO) is important in chordal graphs. It turns out that all the existing chordal graph recognition algorithms and many optimization problems in chordal graphs make use of PEO [48], [53], [81], [89], [116], [118], [124],
[137] and [138]. Fulkerson and Gross [47] characterized these graphs as the Cliapler I Inlroduclion
graphs having a perfect elimination scheme. Dirac [41] showed that these
graphs are the graphs for which every minimal vertex separator induces a
complete subgraph. Buneman [18], Gavril [49] and Walter [149] proved that a
graph is chordal if and only if it is an intersection graph of the subtrees of a
tree. Various other characterizations and results not mentioned here can be
foundin[3], [5], [53]and[108].
Quasi-chordal graphs were introduced by Voloshin in [147] as a
generalization of chordal graphs. The problem of characterizing the class of
quasi-chordal graphs was raised in [147], and independently in [75], where
they called these graphs as good graphs. Recently, Hoang et al. also studied the same problem in [79]. However, no subgraph characterization of quasi-chordal graphs is known [108]. The recognition problem of quasi-chordal graphs was first studied by Voloshin [146], he also proposed an algorithm for this. Later other researchers, Spinrad [128], Hoang [78] and Gorgos et al. [55] improved the time complexity of the recognition algorithm for quasi-chordal graphs. We refer to [3] and [108] for more information on the class of quasi-chordal graphs.
Weakly chordal graphs
As a natural generalization of chordal graphs, Hayward [71] introduced weakly chordal graphs. This class has given rise to a continuous flow of research [9], [II], [16] and [67]. In particular, the tmie complexity for Chanter I Inlroduclion
recognition of the class has steadily improved over the years. Hayward [71]
was first who proposed an 0(/7^) algorithm for weakly chordal graphs. Later
Spinrad improved this algorithm in [129]. Hayward, Hoang and Maffray [67]
introduced the notion of two-pair, which is used in several characterization and
recognition algorithms for weakly chordal graphs. Using this concept they also
reported an algorithm [67] for the optimization problems on weakly chordal graphs. Arikati and Rangan [2] gave an efficient algorithm for finding a two- pair. Spinrad and Sritharan [127] used this to improve the time complexity of weakly chordal graphs. Hayward showed the presence of a separable set of edges, called handle in [69]. This notion was used by Hayward, Spinrad and
Sritharan [68] to give an 0(w ) time recognition algorithm for weakly chordal graphs. In [66] Hayward answered the question of whether there exists a composition scheme that generates exactly the class of weakly chorda! graphs.
Recently Berry et al. in [9] and [13] established a strong structural relationship between chordal and weakly chordal graphs, where they applied variant of
Lekkerkerker and Boland's recognition algorithm for chordal graphs to the class of weakly chordal graphs. This yields a new characterization of weakly chordal graphs, which is not based on two-pair, but rather on the structural properties of the minimal separator of the graph. Various other results and the literature related to weakly chordal graphs can be found in [3], [53] and [108]. Chapter I Introduction
Perfectly orderable graphs and Brittle graphs
In 1981, Chvatal [26] defined the class of perfectly orderable graphs,
motivated by the graph coloring problem. This class of perfect graphs contains chordal graphs, quasi-chordal graphs and various others [3]. In [26] Chvatal asked, whether perfectly orderable graphs can be recognized in polynomial time? Middendorf and Pfeiffer [94] proved that the problem of recognizing perfectly orderable graphs is NP-complete. However, many classes of perfectly orderable graphs such as Brittle graphs, Bipolarizable graphs, /'4-indifference graphs, /'4-comparability graphs, Ps-free weakly chordal graphs and various others [3], have polynomial time recognition algorithms. Since Chvatal's seminal paper [26] appeared in 1984, almost 100 papers on perfectly orderable graphs have been published. For a recent survey, see the chapter -7 by Hoang in [108]. We also refer to [70] for historical details on perfectly orderable graphs.
The class of brittle graphs was shown as one of the first subclass of perfectly orderable graphs by Chvatal [25]. In [74], Hoang and Khouzam proved that brittle graphs are closed under complementation, while substitution does not preserve brittleness. Recognition problem on brittle graphs was first solved in [74]. Later others like, Spinrad and Jonhson [126], Schaffer [120] and
Eschen et al. [42] also solved the same problem and improved time complexity of the recognition algorithm for brittle graphs. No subgraph characterization of brittle graphs is known but there are many known subclasses of brittle graphs
10 Chapter 1 Inlroduction
such as chordal graphs, quasi-chordal graphs, bipolarizable graphs, P^-
indifference graphs, superbrittle and quasibrittle graphs which have subgraph
characterizations. For more information on brittle graphs we refer to [3] and
[108].
Finally other results or informations on perfect graphs or its subclasses,
not mentioned here, can be found in:
(i). "Graph Classes: A survey" by Brandsdadt et al. This survey is
comprehensive with respect to definitions and theorems, citing over 1100
references.
(ii). "Perfect graphs" edited by Ramirez and Reed is a collection of survey
paper's on perfect graphs. It contains almost all main results on perfect graphs
up to 2000.
(iii). "Classes of perfect graphs" by Stefan Hougardy [80] is a useful paper, it lists for each ordered pair {c\, ci) of 120 classes of perfect graphs, whether the class c\ is a subset of the class cj, and if this is not the case, provides and example of a graph which is in C\ but not in cj.
From online point of view, Vashek Chvatal maintains an exhaustive website [24], which is the current pointer to a frequently on-line updated bibliography of perfect graphs. Another online tool for checking the class containment for a larger set of more than 200 graph classes has been developed by Rostock university [150]. Chapter 1 Introduction
Spectral graph theory is the study of the spectra of certain matrices
defined on a given graph, including the adjacency matrix, the laplacian matrix
and other related matrices [36]. The most common matrix investigated has been
the adjacency matrix. Useful information about the graph can be obtained from
the spectra of the matrix. There are also important applications to other fields
such as Chemistry [140]. Some of the common problem studied in the graph
spectra are study of cospectral graphs, bounds on eigenvalues, energy of graphs
and various others [36]. Graph spectra have been studied extensively in the
literature. Here we present only brief information on cospectral graphs and
energy of graphs. We refer to several books such as [22], [34], [36] and [37] for
a more thorough discussion and list of references to original papers.
Cospectral graphs have been studied from the beginning of the development of (he theory of graph spectra [36]. Collatz and Sinogowitz [32] were the first to report that non-isomorphic graphs can have same set of eigenvalues. They gave an example of non-isomorphic trees which are cospectral on 8 vertices. Sridharan and Balaji [133] studied the problem of cospectral graphs for sc chordal graphs. For various other results related to cospectral graphs we refer to [4], [34], [36] and [37].
The energy £ of a graph is the sum of the absolute values of its eigenvalues. This concept was introduced by Gutman [60], in connection to the bO called total /-- electron energy. For details on the general theory of the total n- electron energy, as well as its chemical application, we refer to [60]. Chapter I Introduction
Considerable work and new concept related to energy has been developed since
the first result reported in [60]. Using the arithmetic/ geometric mean inequality
and the nonnegativity of the variance of the nonnegative numbers, McClleand
[91] gave the general bound for the energy. There are numerous other bounds
[60], [155] and [156], on the energy of a graph usually under the assumption of
some specific structure. Two graphs having the same energy are known as
equi-energetic graphs. This concept was introduced by Ramane et al. in [107].
The construction problem of equi-energetic graphs was discussed in [106]. In
the early papers studying graph energy [60], it was conjectured that among n
vertex graphs the complete graph has the greatest energy. This conjecture was
shown to be false [53]. Since then graphs having energy greater than the energy
of complete graphs on same number of vertices is known as hyper-energetic
graphs. Gutman et al. in [61] has shown that no Huckel graph is hyper- energetic. The construction problem for hyper-energetic graphs was first proposed by Walikar et al. in [148]. Additional results on hyper-energetic graphs were subsequently reported in [136] and [152]. Maximal energy [85] is also an interesting concept related to energy of graphs. In [85], Koolen and
Moulton gave the bound for the maximal energy of graphs. Moreover, they showed that the bound is sharp for strongly regular graphs. Maximal energy bipartite graphs are studied in [84]. Hou et al. studied the same problem for unicyclic graphs in [153]. For a recent survey on graph energy and its related concepts, we refer to [ 17]. Chapter I Introduction
1.2 Outline of the thesis
In this thesis we study some structural and algorithmic aspects of certain subclasses of sc perfect graphs. In chapter 1 we present some definitions, introduce perfect graphs and its various subclasses and give a brief introduction of the outline of the thesis.
In chapter 2, we study sc chordal and sc weakly chordal graphs. We study the relation between the two-pair and sc weakly chordal graphs and report a lower and upper bound for this. Using this bound we propose an algorithm for finding chordless cycle C4 in sc weakly chordal graph which is not chordal. We also study the recognition problem of sc chordal and sc weakly chordal graphs and propose algorithms in both the cases. Using these algorithms we compile the catalogue of sc chordal and sc weakly chordal graphs up to 17 vertices. The construction problem of sc graphs is a fundamental problem in studying sc graphs. We study construction problem for sc chordal and sc weakly chordal graphs and propose algorithms for both the cases.
In chapter 3, we extend our study to the other subclasses of sc perfect graphs, such as sc brittle graphs, sc quasi-chordal graphs and jPj-free sc weakly chordal graphs. One of the important relation between all these 3 classes is that they are subclass of sc perfectly orderable graphs. We study sc brittle and sc- quasi chordal graphs and report an Oin'm) time recognition algorithms for both the cases. Using these algorithms we recognize sc brittle and sc-quasi chordal Chapter 1 Introduction
graphs up to 17 vertices. In section 3.4 we study Ps-free sc weakly chordal
graphs. We show that every F5-free sc weakly chordal graph is perfectly
orderable and every Fs-free sc weakly chordal is charming graph. We propose
an 0{nm) time recognition algorithm for Ps-free weakly chordal and then
catalogue f.^-free sc weakly chordal graphs up to 17 vertices.
The problem of finding a maximum clique, a minimum colouring, a
maximum stable set and a minimum clique cover of a graph is known as
optimization problems. These four problems are NP- hard in general [48]. In
chapter 4, we study these optimization problems for sc perfect graphs and their
subclasses namely, sc chordal graphs, sc weakly chordal graphs, sc quasi-
chordal, sc brittle graphs. In general obtaining the exact value of the chromatic
number of a graph is quite difficult. However we give an upper and lower
bounds for the chromatic number of sc perfect graph. In section 4.3.1 we first
give an 0{n'^m) time algorithm for finding maximum clique and minimum
coloring for sc weakly chordal graphs. Later we also report an 0{{n+m)n) time
algorithm for finding largest stable set and minimum clique cover for sc weakly
chordal graph. In section 4.4 we study optimization algorithms for sc quasi-
chordal and sc brittle graphs. We report an 0(/7^m) time algorithm for finding
the minimum coloring for sc quasi-chordal and sc brittle graphs.
Isomorphism problem is still unsolved in spite of the several attempts made by researchers. The spectrum of a graph is a graph invariant. Because of the importance of the isomorphism and recognition problem of sc graphs, in
15 Chapter I Inlroduclion
particular sc perfect graphs, we have studied spectral properties of sc perfect graphs and its subclasses.
In section 5.2 we obtain result for Interlacing Theorem when the graph is sc. In section 5.3, we show that the class of sc and sc weakly chordal graphs cannot be determined by its spectrum. In section 5.4, we prove that the least positive integer for which there exist non-isomorphic cospectral sc weakly chordal graph is 12. In section 5.5, we show that there exist non-isomorphic equi-energetic sc weakly chordal graphs with 8 vertices and there does not exist any non-isomorphic non-cospectral equi-energetic sc chordal graphs up to 13 vertices. In section 5.6, we obtain an upper bound for the energy of sc graph and later prove that the maximal energy of sc graph is always smaller than the maximal energy of general graph on n vertices. In section 5.7, we first prove that no sc graph is hyper-energetic on « < 8 vertices. Moreover we also show that there does not exist any hyper-energetic sc chordal graphs up to 13 vertices and there exist hyper-energetic sc weakly chordal graphs on 12 vertices.
Finally for the implementation of various algorithms given in chapters
2,3 and 4 we use C++ programming language. For obtaining various results in chapter 5 we use software 'Mathematica 5.0'.
1.3 Basic Definitions and Notations
Definition 1.1. A graph G consists of a finite non-empty set V{G) called the set of vertices together with a prescribed set E{G) of unordered pairs of distinct Chapter I Iniroduciion
vertices of G. The number of vertices and the number of edges of a graph G are denoted by n and m respectively.
Definition 1.2. The degree of a vertex v in G is the number of edges incident with it and it is denoted by d{v). The maximum degree of G is denoted byA{G).
Definition 1.3. The degree sequence of a graph G is the sequence of the degrees of the n vertices of G arranged in non-increasing order and is denoted by di > d2> ... > cl^.
Definition 1.4. A subgraph // of a graph G is a graph having all its vertices and edges in G.
Definition 1.5. Let C be a graph. For any subset H of K(G), the vertex induced subgraph G[H] is the maximum subgraph of G with H as the vertex set.
Definition 1.6. A walk of a graph G is an alternating sequence of vertices and edges V(),ei,v,.e2,V2,...,Vn.i,en,v„, where e, = (v,.,, v,). A walk is closed if vo = v^
A walk is a path if all the vertices (and thus all the edges) are distinct. A path consisting of n vertices is denoted by P^. A closed path is called a cycle. A cycle consisting of A? vertices is denoted by C„. A vertex induced subgraph of a graph G which is a cycle is called an induced cycle of G.
Definition 1.7. A graph is C4-free if it does not have any induced C4.
Definition 1.8. Let G be a graph. For a vertex v of G, the graph G - v is obtained by deleting the vertex v and all the edges incident on v. Chapter 1 Inlrodiiciion
Definition 1.9. A graph is said to be complete if every pair of its vertices are
adjacent. A complete graph with n vertices is denoted by K„. A clique of a
graph is maximal complete subgraph of the graph. A clique C of a graph G is
maximum if there is no other clique of G with more number of vertices than the
number of vertices in C. For a graph G the number of vertices in a maximum
clique ofG is called the clique number of C and it is denoted hy (t){G).
Definition 1.10. Let G a be graph. A subset S of G is stable set (also called an
independent set) if no two vertices of S are adjacent in G. A stable set of G
which has maximum number of vertices in it is called a maximum stable set of
G. The number of vertices in a maximum stable set of G is called the stability
number of G and is denoted hy a{G).
Definition 1.11. A clique cover of order A; of a graph G is a partition of its
vertex set K(G) into k subsets V^, Vj, ..., Vy such that each K, is a clique of G.
A clique cover of G with smallest order is called a minimum clique cover of G.
The order of a minimum clique cover of a graph G is called the clique cover
number of G and is denoted by 6'(G).
Definition 1.12. A proper vertex coloring of a graph G is assigning colors to the vertices of G such that no two adjacent vertices are assigned the same color.
The chromatic number/(G) of a graph G is the minimum number of colors needed to properly color the vertices of G.
IH Chapici I Introduction
Definition 1.13. A graph G is /-perfect if /(//) = subgraph H of G. A graph G is a-perfect if a{H) = 6{H) for every induced subgraph H of G. A graph G is perfect if it is / -perfect (or equivalently it is a -perfect). Definition 1.14. A graph is spUt graph if its vertex set can be partitioned into C and S such that C is a clique of G and 5 is a stable set of G. Definition 1.15. Let G be a graph with n vertices whose vertices are labeled 1,2,...,«. The adjacency matrix is defined as the n x n matrix {a,^) where o^ is given by r- \ if a, J) eE{G) o,j - 0 otherwise Definition 1.16. Two graphs G'and Care isomorphic if there exists a one-to- one correspondence between the vertex sets of G' and G" which preserves the adjacencies of the vertices. Such a one-to-one correspondence is called a vertex isomorphism of G' onto G" and is denoted by ij/. If G' and G" are isomorphic, then we denote it by G' s G". Definition 1.17. The complement G of a graph G has V{G) as its vertex set and two vertices are adjacent in G if and only if they are non-adjacent in G. Definition 1,18. A graph G is self-complementary (sc) if it is isomorphic to its complement G. The number of non-isomorphic sc graphs with n vertices is denoted by 5„. Chapter I Iniroduclion Definition 1.19. A graph G is chordal if it has no chordless cycle of length greater than or equal to 4 and it is weakly chordal if both G and its complement G have no chordless cycle of length greater than or equal to 5. Definition 1.20. A self-complementary graph which is also chordal (respectively, weakly chordal) is called sc chordal graph (respectively sc weakly chordal graph). Definition 1,21. A list of all non-isomorphic sc graphs with n vertices is called the catalogue of sc graphs with n vertices. A list of all non-isomorphic sc chordal graphs with n vertices is called the catalogue of sc chordal graphs with n vertices. Definition 1.22. Two non-adjacent vertices x,y in a graph G form an even pair if every chordless path between them has an even number of edges and called two pair if every chordless path between them has length two. A co pair is a complement of a two pair. Definition 1.23. A hole is an induced cycle with five or more vertices and an antihole is the complement of a hole. Definition 1,24. For two vertices x, v in a graph G, the graph obtained by deleting x and y and adding a new vertex xy adjacent to precisely those vertices of G - X - y which were adjacent to at least one of x, y in G is called contraction of two vertices. 20 Chaplcr I Iniroduclion Definition 1.25. Let G be a graph. A vertex x is simplicial if its neighborhood A'(x) induces a complete subgraph and it is co-simpHcial if its non-neighbors form an independent subset of vertices. Definition 1.26. Let G be a graph with an induced P^ [a, b, c, d\, vertices b, c are called midpoints of the P^ and a, d are called endpoints of the P4. A vertex of a graph G is no-mid (respectively, no-end) if it is not the midpoint (respectively, endpoint) of any PA in G. Set of no-mid vertices is called no-mid set, similarly set of no-end vertex is called no-end set. Definition 1.27. A graph G is brittle if each induced subgraph H of G contains a vertex that is not a midpoint of any P^ or not an endpoint of any A. A self- complementary graph which is also brittle is known as sc brittle graph. Definition 1.28. A graph is quasi-chordal if each of its induced subgraphs contains a simplicial or co-simplicial vertex. A self-complementary graph which is also quasi-chordal is known as sc quasi-chordal graph. Definition 1.29. A sc weakly chordal graph is called Pj-free (Ps-free) sc weakly chordal if it has no induced subgraph isomorphic to P5 (P5, respectively). Definition 1.30. The eigenvalues of a graph G are the eigenvalues of its adjacency matrix. The spectrum of a graph G is the multiset of eigenvalues with their multiplicities. ClianlcT I Inlroduclioii Deflnition 1.31. Two graphs are cospectral whenever they have the same spectrum. A graph G is determined by its spectrum (DS) if every graph cospectral with G is isomorphic to G. Definition 1.32. Let G be a graph with n vertices and /l, > Aj - — - K ^^^ ^he eigenvalues of G. Then the energy of G, denoted by E{G) is E(G) = ^| A< |. Definition 1.32. Two graphs Gj and Gj with E{G\) - EiGj) are said to be equi-energelic graphs. If for a graph G, E{G) > 2{n-\), then G is said to be hyper-energetic graphs. For other definitions we refer to [ 108] and [ 154]. Notations In this thesis a denotes the end of a proof Let r be a real number. The greatest integer less than or equal to r and the smallest integer greater than or equal to r are denoted by \_r\ and| r | respectively. 22 CHAPTER TWO Cliaplcr 2 On tc cliordal iind sc weakly chordal sruphs Chapter-2 On sc chordal and sc weakly chordal sraphs 2.1 Introduction Study of any graph class includes characterization, recognition, counting the number of graphs i.e. cataloging and construction of graphs. In this chapter we design a recognition algorithm based on the characterization of sc chordal graphs given by Sridharan and Balaji [134]. Using this algorithm, we count the number of sc chorda! graphs up to 17 vertices. We further give an algorithm for the construction of sc chordal graphs. Sc weakly chordal graph is one of the natural generalization of sc chordal graphs. We study all the above four mentioned problems for sc weakly chordal graphs. In section 2.2, we briefly discuss some basic results used in this chapter. Section 2.3 deals with sc weakly chordal graphs and two-pairs. In section 2.4 we present recognition algorithms for sc chordal graphs and sc weakly chordal graphs. Cataloging is done in section 2.5. Finally in the last section 2.6, construction of sc chordal and sc weakly chordal graphs is discussed. 2.2 Basic Results For obtaining various results in this chapter for sc chordal and sc weakly chordal graphs we use the following results available in the literature. 23 Chapter 2 On sc chordal and sc weakly chordal eraphs Theorem 2.1 [71|. Every chordal graph is weakly chordal graph but converse need not to be true. The graph given in Figure-2.I is weakly chordal but not chordal. Vj# * V, Figure-2.1 The following result was given by Gibbs [52] for the number of disjoint f4's in sc graph. Theorem 2.2 [521. A sc graph on /? = 4/? or n = 4p+\ vertices contains p disjoint induced A's. The following Theorem was given by Hayward, Hoang and Maffray [67], which characterizes weakly chordal graph. Two-pair Theorem 2.3 |67|. A graph is weakly chordal if and only if each induced subgraph is either a clique or contains a two-pair of the subgraph. We use two-pair Theorem for the recognition of sc weakly chordal graphs in section-2.4 2.3 Sc weakly chordal graphs and two-pairs Two-pair is an important concept in weakly chordal graphs. In this section we study the relation of two-pair with sc weakly chordal graphs and later give the lower and upper bounds for the number of two-pairs in sc weakly chordal graphs. We also give an algorithm for producing chordless cycle C4 24 Chaplcr 2 On sc chorjal and sc weakly chorda} graphs using the two-pair concept in sc weakly chordal graphs. For obtaining bounds on two-pair for sc weakly chordal graphs we need the following result which was reported by Berry in [12]. Theorem 2.4 [12|. A non-clique weakly chordal graph has at least two non- adjacent co-pairs. The following Corollary follows from the above Theorem. Corollary 2.5. A non-clique weakly chordal graph has IKi as induced subgraph. Proof. Since a co-pair is the complement of a two-pair which indeed fonns an edge {K2) of a graph. By the above Theorem-2.4, there exist another non- adjacent co-pair if the graph is non-clique weakly chordal graph. Hence these two non-adjacent edges (K-l) form an induced subgraph (^Ki) in a non-clique weakly chordal graph, D Now for the class of sc weakly chordal graphs we have the following result. Theorem 2.6. Let G be a sc weakly chordal graph but not chordal then it has at least 2 two-pairs. Proof. Let G be a sc weakly chordal graph that is not chordal, this sc weakly chordal graph is non-clique since a sc weakly chordal graph has /7(/7-l)/4 edges. Now by Theorem-2.4 in this sc weakly chordal graph there exists at least two non-adjacent co-pairs. These two non-adjacent co-pairs become two 25 Chapter 2 ^ On sc chordal and sc weakly chordal graphs different two-pairs in G (since G and G are isomorphic), thus in G there also exists at least 2 two-pairs. Hence the Theorem, D The following result gives lower and upper bounds for the number of two-pairs in sc weakly chordal graph. Theorem 2.7. Let G be a sc weakly chordal graph but not chordal with n = 4p or r? ^ 4/7+] vertices then the number of two-pairs in G satisfy 2 < number of two-pairs < ( p). 4 Proof. First we prove for upper bound, since G is sc thus it has n{n-l)/4 edges and same number of non-adjacent vertices. Now the maximum number of two- pairs that exists in G is n{n~\)/4. From Theorem-2.2 we know that in a sc graph with n = 4/7 or n = 4p+\ vertices there exists/? disjoint induced P4 in G. Let [a, b, c, d\ be an induced P4 in G, then the vertices a, c and b, d both have chordless path of ^ ^ ^ ^ length 2 between them, thus they may become two-pair in G. While non- adjacent vertices a and c/have chordless path of length 3 in induced P4, so there will be a path of length three in G also. Therefore the non-adjacent pair {a, d] cannot become a two-pair either in P^^ or in G. Thus in G, at least p pair of non- adjacent vertices cannot become two-pair among n{n-\)IA non-adjacent vertices. Hence the upper bound. The lower bound follows from Theorem-2.4n The following result shows the behavior of two-pair in sc weakly chordal graphs. 26 Chapter 2 On sc chordal and sc weiiklv chordal fjruphs Theorem 2.8. Let G be a sc weakly chordal graph but not chordal, then there will be 2 disjoint two-pairs {xi,;/|} and {xj, yj} of G such that the vertices Xi, y],X2 and 72 will induce a chordless cycle of length 4. Proof. Let G be a sc weakly chordal graph that is not chordal, then by Theorem-2.6, there will be at least 2 two-pairs. Let {xi,_yi} and {x2, yi) be these 2 two-pairs in G. To prove that the vertices X\, yi, Xj and yj induce a chordless cycle C4, we consider only induced subgraphs on 4 vertices in G. Since there are only 11 graphs on 4 vertices and we check one by one whether any of these 11 graphs can contain 2 disjoint two-pairs. Case (1)- If the graph induced by 4 vertices is disconnected with edges m < 3, then there will be no 2 two-pairs as it can be seen in the following figure-2.2(i- V). • • • • • (i) (ii) (iii) ( iv ) ( V ) Figure-2.2 However graph in (v) contains only one two-pair. Thus in this case no 2 two- pairs exists and no chordless cycle C4 is obtained. Case (2)- If the graph induced by 4 vertices is connected having edges m > 4, then we have graphs as given in figure-2.2(vi and vii) and these graphs do not contain 2 two-pairs and no chordless cycle C4 is produced. ( vi ) ( vii ) Figure -2.2 27 Chapter 2 On sc chorJal and sc ueakly chorJal graphs Case (3)- Now we consider the following connected graphs induced by 4 vertices as shown in figure-2.2(viii & ix). In both graphs although 2 two-pairs exist but they have one vertex common between the 2 two-pairs hence they are neither disjoint nor produce a chordless cycle C4. ( viii) (ix ) Figure -2.2 Case (4)- Now we have a connected graph induced by 4 vertices having 3 edges. These vertices form 2 two-pairs with no vertex common then the only possible graph is P4. • • • • Figure-2.2 This is clearly not a chordless cycle C4. Case (5)- The remaining graph on 4 vertices among the 11 graphs is now clearly the chordless cycle C4, Figure -2.2 (xi) where the 2 two-pairs {X|,>'|} and {x2, V2} form chordless cycle C4. Hence the result, a Since a sc weakly chordal graph which is not chordal contains chordless cycle C4, hence for finding this cycle using the two-pair concept, we develop an algorithm based on Theorem-2.8. Moreover in the process of finding chordless C4 in sc weakly chordal graphs the proposed algorithm first requires to find a 28 Chapter 2 On sc chorja/ and sc weakly chordal graphs two-pair in the input graph, for this we borrow the idea from [2] for finding a two-pair in the input graph. In [2] Arikati and Rangan provide a method for finding a two-pair in a graph in 0{mn) time. However, the fastest algorithm for finding a two-pair in a graph runs in 0(w^ '*^) time [86]. But it uses theoretically fast matrix multiplication algorithm of [105], making it relatively impractical. The following algorithm-2.1 works as follows: first it finds arbitrary non-adjacent pair of vertices {x, y}. If no non-adjacent pair of vertices exists then obviously no two-pair exists (but this never happens in the case of sc graphs). After finding a non-adjacent pair {x, y], it computes there open neighborhood A^(A-) and N(y) also it computes N{x) H N{y). Now the algorithm finds /?v i.e. remaining vertices in the graph from V{G) - {N{x) C\ N{y)). Now if the vertices x and v lie in different components of R^, then the algorithm returns that {x,y} is a two-pair and if not then it returns {x,y} is not a two-pair. Algorith m 2.1: An Algorithm for recognizing a two-pair in a sc graph. Input: A sc graph G given by adjacency list. Output: A two-pair {x,y} if it exists. Begin While ( G has non-adjacent pair {x,y}) compute N{x), N{y) and N{x) n N{y). find y{G) - [ N(x) nN(y)] = R, (let). if in /?v X &y are disconnected, return {.x, v} is a two-pair 29 Chapter 2 On sc chordal and sc weakly chordal ffraphs else return {JC, y} is not a two-pair end if endwhile End. The above algorithm-2.1 is used as a subroutine in various other algorithms which we discuss later in other part of the thesis. Now we give an algorithm for finding a chordless cycle C4 in sc weakly chordal graph. The following algorithm produces a chordless cycle C4 using two-pair in the given sc weakly chordal graph. Algorithm2.2: An Algorithm for finding chordless C4 in sc weakly chordal graph. Input: A sc weakly chordal graph but not chordal. Output: A chordless cycle C4. Step 1: Generate list of all possible two-pairs (using algorithm-2.1) and list of all induced ^4. Step 2: Pick 2 two-pairs till there exists 2 two-pairs. Step 3: If there is any vertex common between them then go to step 2. Step 4: If they lie on any induced P4, go to step 2, Step 5: Two-pairs will produce chordless cycle C4. (by Theorem-2.8). End. .30 Cliaplcr 2 On sc chordal and sc weakly chordal graphs 2.4 Recognition of sc chordal and sc weakly chordal graphs Recognizing chordal graph has been extensively studied in literature [55]. Fulkerson and Gross [47] recognized chordal graphs using the following iterative procedure; repeatedly locate a simplicial vertex and eliminate it from the graph, until no vertices remain (and the graph is chordal) or at some stage no simplicial vertex exists (and the graph is not chordal). Their procedure take 0(/7'') time, while later some other researchers [89], [116] and [118] recognized chordal graphs in h'near time by using LexBFS and MCS on perfect elimination ordering. For the class of sc chordal graphs Sridharan and Balaji recognized chordal graphs using the characterization given in [131] and [134] and catalogued sc chordal graph up to 13 vertices. In this section we also recognize sc chordal graph using a characterization given by Sridharan and Balaji [134], and later improve the catalogue of sc chordal graphs up to 17 vertices. The following result was given by Hammer and Simone [63] which characterizes split graphs in temis of degree sequence. Theorem 2.9 [ 63). Let G be a graph with degree sequence d\>d2>... > (/„ . Let M = max{/: J, >/-l,l < / <«} then G is a split graph iff x,t',£/, =A For sc graphs Sridharan and Balaji [134] proved that the value of M=2p for n = 4p and M = 2p+\ for n = 4p+\. Further using Theorem-2.9 and the value of M for sc graph they obtained the following result for sc split graphs. 31 C'haplcr 2 On sc chordal and sc weakly chordal graphs Theorem 2.10 |134|. Let G be a sc graph with degree sequence J| >d2 >...>d„. Then G is split graph iff "^.fd, =6p^ -2p for/? = 4/? and Yj^!\cr^(>P^ for« = 4/7+1. Foldes and Hammer [45] gave the following result for a split graph to be chordal. Theorem 2.11 [451. Let G be a graph. Then G is a split graph iff both G and G are chordal. When G is sc graph, the following Corollary is immediate from the above Theorem. Corollary 2.12. Let G be a sc graph. Then G is split graph iff G is chordal graph. From the above Corollary it is clear that sc split graph class and sc chordal graph class are equivalent to each other therefore the following Theorem is natural for sc chordal graphs. Theorem 2.13 {134|. Let G be a sc graph with degree sequence J, >d2 ^..-^o',,- Then Gis chordal graph iff ^.'jt/. =6p^ -2p for/7 = 4p and XM^-^^Z'^ forn = 4p+ I. Proof. Follows from Theorem-2.10 and Corollary-2.12. a On the basis of the preceding Theorem we propose an algorithm-2.3 for recognizing sc chordal graph in linear time. The algorithm works as follows: 32 Chapter 2 Un sc chorjal and s<: weakly chordal graphs since we input a sc graph with the given degree sequence, it immediately arranges the degree sequence in non-increasing order. Now if the input graph has Ap vertices then it checks for the condition, ad. = 6p^-2p. Otherwise it /=i ~i> •> checks for ^:c/, = 6/?' when the number of vertices is 4p+\. If the conditions are satisfied it produces "yes" i.e. the input graph is sc chordal, else it produces "No". The following algorithm recognizes whether a given sc graph with 4p or 4p+\ vertices is chordal or not? Algorithm 2.3: An Algorithm for recognizing sc chordal graph. Input: A sc graph with 4p or 4p+\ vertices with degree sequence. Output: "yes" if sc graph is chordal else "No". Begin arrange the degrees in non- increasing order for i = 1 to 2p Sum = Sum + cl, If ( no of vertices = 4p) then If ( Sum = 6p^ - 2p ) then return " Yes" else return " No" endif else if (sum = bp") then retum "yes" else return "No" endif endif Endfor End. 33 Chapicr 2 On sc choidal and sc weakly chordal graphs The complexity of algorithm-2.3 can be computed as follows. Complexity. Since the degree sequence of sc graph is given and all the degrees are integers, so it takes 0(/7) time for sorting the degrees on n vertices of G. Now, to get the sum of degrees up to 2p vertices it requires 0(«) time. Moreover to check the IF Condition 0(«) time is required . Hence the overall time complexity is 0(/7). The correctness of the algorithm-2.3 is established in the following Theorem. Theorem 2.14. Algorithm-2.3 checks whether an input sc graph is chordal or not correctly. Proof. Follows from Theorem-2.13. D Although a graph can be recognized to be chordal or not in linear time by the algorithms given in [89] and [116] using the simplicial vertex concept. But recognizing that a chordal graph is sc or not appears to be difficult. To illustrate algorithm-2.3 we consider the sc graphs G\ and Gj on 8 vertices as shown in figures-2.3(a) and 2.3(b) respectively. (b) (a) Figure-2.3 Let the sc graph G\ be the input graph to algorithm-2.3. The algorithm first arranges its degrees in non-increasing order as 34 Chapter 2 On .vc chori/al and sc weiiklv chordal f d{v^)>cl{v^)>d{v^^)>d{v^)>d{v,)>d{v^)>d{v,)>d{v^), (which is 6 > 6 >4>4>3>3>I>1). Since the graph d has 8 vertices i.e. 4p = 8 orp = 2, therefore the algorithm checks for the condition (sum = 6p^ - 2p) and it finds, Sum = iW, = (6+6+4+4) = 20 and dp' - Ip = 6(2)^ - 2(2) = 20. Since Sum (id,) = 20 = 6p~ - 2p algorithm 2.3 decides that the graph C| is a chordal graph. Now let the sc graph Gj shown in figure-2.3(b) be input graph to the algorithm-2.3. The algorithm first arranges its degrees in non-increasing order as t/(Vj() >d{v^)>divj>d{v^)>d{v^)>«'(v,)>div^)>d{v,), (which is 5 > 5>4>4>3>3>2>2). Again for/7 = 2 algorithm checks for condition (sum = 6/;^ - 2p), now it finds Sum = iW = (5+5+4+4) = 18 and 6/;^ - 2p = 6(2)^ - 1=1 2p -, 2(2) = 20. Thus Sum (id-) = 18 # 6p'' - 2p, algorithm 2.3 decides the graph i=\ Gj is not a chordal graph. Similarly algorithm-2.3 can also check for 4/^+1 vertices also. The problem of recognizing weakly chordal graph has been studied in the context of finding chordless cycles of length greater than or equal to 5. Hayward [71] proposed an 0(/7 ) time sequential algorithm for detecting chordless cycles of length greater than or equal to k. This algorithm can be used to recognize whether a graph G is weakly chordal by checking the presence of chordless cycle of length greater than or equal to 5 in G and then in its 35 Chapter 2 On sc chordal and sc weakly chordal praphs complement G; this in turn leads to an O(n^) time recognition algorithm for weakly chordal graphs. Hayward's result was improved to 0{n*^^^) by Spinrad's hole-finding procedure [129]. Arikati and Rangan [2] provided a method for finding a two-pair in a graph in 0{mn) time. Spinrad and Sritharan [127] used this to provide a sequential algorithm for recognizing weakly chordal graphs in 0{n'm) time. Essentially the algorithm for recognizing weakly chordal graphs given by Spinrad and Sritharan rely on the following fact. Edge Addition Theorem 2.15 [127]. Let G be a graph and let {x, y) be a two-pair in G. Let G' be the graph obtained from G by adding an edge between X and V. Then G is weakly chordal if and only if G' is weakly chordal. They also proved that the process of adding an edge between the vertices contained in a two-pair does not create or destroy either hole or antihole. Other recognition algorithms for weakly chordal graphs can be found in [13], [68] and [97]. Now we are in a position to discuss an algorithm for recognizing sc weakly chordal graphs. It proceeds by computing a sequence of graphs, starting with the given sc graph as input, it finds a two-pair (if it exists) this can be done by using algorithm-2.1. After finding a two-pair it joins an edge between these two non-adjacent vertices. The algorithm repeatedly finds two-pairs and adds edges between them i.e. the graphs which are obtained later in the sequence have strictly more edges than the graphs those obtained earlier in the sequence. 36 Chapter 2 On sc chonial cinii sc neaklv chorda) craphs If all the non-adjacent vertices are now adjacent i.e. the last graph becomes clique, then the algorithm returns that the input sc graph is weakly chordal. Otherwise if, at any stage of the above procedure it does not find any two-pair and there are still remaining non-adjacent pair of the vertices then it returns that the input graph is not sc weakly chordal graph. The algorithm is as follows. Algorithm 2.4: An Algorithm for recognizing sc weakly chordal graph. Input: A sc graph G given by adjacency list. Output: "yes" if G is sc weakly chordal else "no". Begin While ( G has two-pair) do forx <-l to n-\ do for each y i Adj(x) do status <- find two-pair {x. y] if status = True then join edge between x &.y go to while loop endif endfor endfor endwhile if G is clique then return ( "yes " ) else return ( "no " ) endif t:nd. 37 Chapter 2 On sc chordal and sc weakly chordal eraphs The time complexity of aIgorithm-2.4 can be computed as follows. Complexity. Since we are beginning non-adjacent pair {x, y) we can determine if {x, y} is a two-pair in 0(m) time by checking if there is a path between x and y in the graph G - {N{x) n N{y)). So for all non-adjacent pairs the algorithm takes time order ^ | Nonadj(x) |.w = w ^ | Nonadj(x) | = w./r; = 0{m~), as a worst case. In while loop the worst case is that we find only one two-pair at each iteration. Hence the number of times the two-pair can be found in G is equal to the number of edges present in G i.e. m. Therefore the maximum number of times the while loop is executed is m. Hence the total complexity of algorithm is 0{in^.m) = 0{m). The following result shows the validity of the algorithm-2.4 Thcorem2.16. Algorithm-2.4 checks whether an input sc graph is weakly chordal or not correctly. Proof. Let us input a sc graph, in algorithm-2.4, it repeatedly finds two-pair {A-, ,V} and joins an edge between x and>' by edge addition Theorem-2.15. This process will not destroy the property of weakly chordal graph. In this way if the final graph is clique, then the input graph is sc weakly chordal graph. This proves the validity of the algorithm, a To illustrate algorithm-2.4, we consider two sc graphs G\ and G2 on 8 vertices, as shown in figure-2.4(a) and figure-2.4(b)respectively . Let graph Gi be the input to algorithm-2.4. The algorithm first finds pair of 38 Chanter 2 On sc chordul and sc weakly chordal graphs Figure-2.4 non-adjacent vertices {v2, V3} as one of the two-pair in G\ (this is done by using algorithm-2.1). Now algorithm adds an edge between these two vertices \>2 & V3 and updates the adjacency list of the input graph G]. After updating the list it again finds {v2, vg} as two-pair and adds an edge between these two vertices then it updates the adjacency list of G\. Repeating the above procedure the algorithm finds {v,, vj}, {v,, v^}, {v,, V3}, {vj, v^}, {vj, vg}, {vy, V4}, {vy, v^}, {VA, Vi), {V4, V7}, {V4, Vft), {vs, V(,} and {V5, v^} as two-pairs at successive stages and joins an edge between them, and update the adjacency list of G\ at each stage. Since all the non-adjacent pair of vertices are joined by the edges therefore the input graph G] becomes clique and algorithm-2.4 decides that G| is sc weakly chordal graph. Let graph Gj be the input to algorithm-2.4. The algorithm first finds {vi, V3} as one of the two-pair and adds an edge between them, then it updates the adjacency list of Gj. Now it finds at second and third stages pair of non- adjacent vertices {V3, V4} and {1^4, V(,} as two-pairs. After adding the respective edges and updating the adjacency list of G2 at each stage, it does not find 39 Chapter 2 On sc choidal and sc weakly chorda! firaphs further any two-pair in G2 and the graph Gj does not become a clique. Therefore algorithm terminates and produces the output that the graph G2 is not sc weakly chordal graph. However the other reason for the graph G2 is not weakly chordal is that the vertices {V2, V5, vg, V6, V7} form a chordless cycle C5. However the vertices {1^2, I's), {^2, V(,}, {V5, Vfy}, {V5, V7} and (V7, v^} in C5 can never become two-pair in Gj- 2.5 Catalogue Compilation of sc chordal and sc weakly chordal graphs General progress in the construction and cataloging of sc graph has been slow. Catalogue of sc graphs with 8 vertices were compiled by Alter [1], Faradzhev [43], Morris [95] and Venkatachalam [145]. Faradzhev [43] and Morris [95] also compiled the catalogue of sc graphs with 9 vertices. Faradzhev [43] and Kropar et al. [87] obtained the catalogue of sc graphs with 12 vertices. Kropar and Read in [87] observed that obtaining the catalogue of sc graphs with more than 12 vertices was quite difficult. However recently Gordon Royal [56] provided the catalogue of sc graphs upto ! 7 vertices. In [133] Sridharan and Balaji catalogued sc chordal graphs upto 13 vertices using existing catalogue of sc graphs. The following table-2.1 shows the number of non-isomorphic sc chordal graphs upto 13 vertices as reported in [133]. 40 Chapter 2 On sc chorda/ and sc Heakh chordal eraphs Number of vertices n 4 5 8 9 12 13 Nuniberofsc graphs 1 2 10 36 720 5600 Number ofnon-isomorphic sc chordal graphs 1 1 3 3 16 16 Table-2.1 We improve their catalogue of sc chordal graphs from 13 vertices to 17 vertices. Using algorithm-2.3, we recognize chordal graphs form the catalogue of sc graphs upto 17 vertices and find that there exist 218 non-isomorphic sc chordal graphs on 16 and 17 vertices each. The improved catalogue of sc chordal graphs upto 17 vertices can be seen in the following table-2.2. Number of vertices n 4 5 8 9 12 13 16 17 Number of sc graphs 1 2 10 36 720 5600 703760 11220000 Number ofnon-isomorphic sc chordal 1 1 3 3 16 16 218 218 graphs Table-2.2 With the help of aIgorithm-2.4, we compile the catalogue of sc weakly chordal graphs with at most 17 vertices from the available catalogue of sc graphs. There is only one sc weakly chordal graph on 4 vertices and one on 5 vertices. Both sc weakly chordal graphs on 4 and 5 vertices can be seen in figure-2.5(a) and figure-2.5(b) respectively. (a) (b) Figurc-2.5 Cliapier 2 On sc chorda! and sc weakly chordal graphs There are 7 non-isomorphic sc weakly chordal graphs with 8 vertices. They are shown in figure-2.6 in which graph (a), graph (b) and graph (c) are sc chorda! while graph (d), graph (e), graph (f) and graph (g) are sc weakly chordal but not chordal. (a) (c) (0 42 (Mmi2- On sc chorda! and sc weukly chordal graphs There are 7 non-isomorphic sc weakly chordal graphs with 9 vertices. They are shown in figure-2.7 in which graph (a), graph (b) and graph (c) are sc chordal while graph (d), graph (e), graph (f) and graph (g) are sc weakly chordal but not chordal. (a) (d) (c) (0 43 Ciiapler 2 On sc chordal and sc weakly chordal graphs For more than 9 vertices, as the number of non-isomorphic sc graphs increases similarly number of non-isomorphic sc weakly chordal graphs also increases rapidly which can be seen in following table- 2.3. Number of vertices n 4 5 8 9 12 13 16 17 Number ofsc graphs 1 2 10 36 720 5600 703760 11220000 Number of non-isomorphic sc weakly 1 1 7 7 162 162 14510 14510 chordal graphs Table-2.3 2.6 Construction ofsc chordal and sc weakly chordal graphs The construction problem of sc graphs is a fundamental problem in studying sc graphs. The construction of sc graphs by mean of complementing pennutation was considered in [113] and [119]. In [151] Jin and Wong proposed the decomposition method for the construction of sc graphs. Gibbs [52] has given algorithms for the construction ofsc graphs with 4/? and 4/7+1 vertices. Sridharan and Balaji [132] also gave algorithms for the construction of sc chordal graphs with 4p and 4p+\ vertices by modifying the algorithm as discussed in [52]. In this section we propose a different approach for the construction ofsc chordal graph using the concept of two-pair. Fulkerson and Gross [47] gave the following classical construction scheme for chordal graphs, which works as follows: "Start with a graph Go with no vertices and repeatedly add a vertex Vj 44 s9 Chapter 2 On sc chordal and sc weakly chordal graphs to Gj.i to create the graph Gj such that Vj is not the middle vertex of any ^3 of G;\ This construction scheme for chordal graphs is known as a vertex addition scheme. Now for obtaining the different method for the construction of sc chordal graphs we need the following result which was given by Berry et al. [10]. Theorem 2.17 1101. Let G\ be a chordal graph and [x, y) be a pair of non- adjacent vertices of G\. Let Gj be the graph obtained from G\ by adding edge xy\ then Gj is chordal iff {x,y} is a two-pair of G|. Using Theorem-2.17, We propose the following algorithm-2.5 for the construction of sc chordal graph with the given degree sequence (J, > Jj >... >dj. We start with a graph (G,) having all n isolated vertices and repeatedly add an edge between the non-adjacent vertices (x axxdy) to create the graph (G,+i) such that {x, v} is a two-pair in G,. Algorithm-2.5 repeats this process until required degree sequence is obtained. The following algorithm constructs sc chordal graph with the given degree sequence on ;? vertices. AIgorithm-2.5: An Algorithm for the construction ofsc chordal graph. Input: A degree sequence {d^ >d2 >...>dj ofsc graph. Output; A sc chordal graph with degree sequence (c/, > c/, > ... ^ t/„). Step-1: Start with a graph Gi with n vertices and no edges. 45 Chapter 2 On sc chordal and sc weakly chordal graphs Step-2: Add an edge between two non-adjacent vertices {x and y) of G\, to. obtain graph Gj such that {x,y} is a two-pair in G\. Step-3: Repeat the process of step-2, until we get a graph with degree sequence ((5^1 >d2 >...>dj. End. The following Theorem states that a graph G with n vertices is a sc chordal iff it can be constructed by algorithm-2.5 Theorem 2.18. (i). The constructed graph by algorithm-2.5 is a sc chordal graph with (a', > (T', >... > dj degree sequence. (ii). Every sc chordal graph with (d^ td^ >...>d„) degree sequence on n vertices can be constructed by algorithm-2.5. Proof (i). The algorithm-2.5 takes degree sequence (c/, >d2 >...>dj of sc graphs as input and it construct chordal graphs from empty graphs using Theorem-2.17, which ensures chordality. Also from Theorem-2.13, no other graph is possible with the same degree sequence. (ii). Follows from Theorem- 2.17. D We illustrate algorithm-2.5 by constructing a sc chordal graph on 12 vertices. Suppose we have to obtain a sc chordal graph on 12 vertices with degree sequence (10,10,8,8,6,6,5,5,3,3,1,1). Algorithm-2.5 starts with an empty graph Gi on 12 vertices {V], vj, vy, v^, V5, V(„ v?, v^, vy, V|o, vn, V]2} as shown in figure-2.8(a). The overall procedure of the construction of sc chordal graph 46 Chapter 2 On sc chordal and sc weakly chordal graphs with the given degree sequence (10,10,8,8,6,6,5,5,3,3,1,1) is obtained as given below and illustrated in figure-2.8 (i). The graph Gj as shown in figure-2.8(b) can be easily obtained, since in G\ there is no edge and in the given degree sequence there is a vertex of degree 10. So we select any vertex ( let it be V|) arbitrarily to give it adjacency 10 by adding edges e\, ej, e^, CA, e^, e^, ej, e^, e<), eio from vertex v, to vj, vj, V4, V5, V6, V7, vg, V9, vio and vn respectively. In this way we get graph Gj as shown in figure-2.8(b). Obviously there is no violation of step-2 of algorithm-2.5 by adding these edges in this way. (ii). In the given degree sequence we have another vertex of degree 10. Let this vertex be vi- Since vj already got 1 adjacency as can be seen in flgure-2.(b). So we give it 9 more adjacencies by adding edges en, e\2, e^, e^, ^15, ei6, ep, eis and ei9 from vertex vi to V3, V4, V5, v^, V7, v^, vg, vio, V12 respectively. In this way we obtain graph G3 as shown in figure-2.8(c). From figure-2.8(b) it is clear that all the non-adjacent pairs i.e. {v2, vy), {v2,V4}, {v':, V5}, {vj, Vf,}, {v2, V7}, {vj, ^s}, b'l, I'g}, {v2, ^'10} and {vi, V12} are two-pairs in Gj. So there is no violation of step-2 of algorithm-2.5. (iii). The next lowest degree in the given degree sequence is 8. Let this vertex be V3. Since vertex vy has already got 2 adjacencies as can be seen in figure- 2.8(c). Therefore we give it 6 more adjacencies by adding the edges 620, ^21, £22, 6?23, ^24 and ^25 from vertex i'^ to V4, V5, v,,, V7, v^ and vq respectively. Doing in this way we obtain graph G4 as shown in figure-2.8(d). From figure-2.8(c) it is 47 Chapter 2 On sc cbordal and sc weakly chordal graphs clear that all the non-adjacent pairs i.e. {V3, V4}, {V3, V5}, {V3, v^}, {V3, V7}, {vj, vg} and {V3, V9} are two-pairs in the graph G3. So there is again no violation of step-2 of algorithm-2.5. (iv). In the given degree sequence there is another vertex of degree 8 (let it be V4). Since vertex V4 has already 3 adjacencies as can be seen in figure-2.8(d). So we add edges (?26, an-, ejn, ejq and eyq from vertex v^ to V5, Vfi, V7, vg and Vio respectively. In this way we obtain graph G5 as shown in rigure-2.8(e). From figure-2.8(d) again it is clear that all non-adjacent pairs i.e. {V4, vs}, {V4, Vf,}, {^4, V7}, {V4, Vfi} and {V4, V|o} are two-pairs in the graph G4. (v). Now the next lowest degree in the given degree sequence is 6. Let this vertex be V5. From flgure-2.8(e) it is clear that vertex V5 has already adjacency 4, so we have to give only 2 more adjacencies in the form of edges e^] and ^32 from vertex V5 to Vf, and v-, respectively. Thus we obtain graph Gf, in this way as shown in figure-2.8(f)- From figure-2.8(e) it is clear that non-adjacent pair {V5, Vft} and {vfi, vy} are two-pairs in the graph G5. (vi). In the given degree sequence another vertex is of degree 6 (let it be V(,). Since vertex v^, already got 5 adjacency, which can be seen in graph G^. So we have to give only 1 adjacency in the form of adding edge 033 from vertex V(, to vg. In this way we obtain graph G7 as shown in figure-2.8(g). From graph G(, it is clear that the non-adjacent pair {v^, v^} is a two-pair. So adding edge between them is allowed. 48 Chapter 2 On ic chorda! and sc weakly chordal sraphs As the graph d has degree sequence (10,10,8,8,6,6,5,5,3,3>1,1) which is the same degree sequence given in the input of algorithm-2.5. Thus algorithm stops here. Since there is no violation of step-2 at any stage of construction, therefore the constructed graph G7 is sc chordal on 12 vertices with degree sequence (10,10,8,8,6,6,5,5,3,3,1,1). V5 Vu V7 v? V4 • •v, V, • (a) G, (c) (b) Figurc-2.8 (continue) 49 Chapter 2 On sc chordal and sc weakly chorda! praphs G, (d) (e) G, (0 (g) Figurc-2.8 50 Chapter 2 On sc chorda! and sc weakly chordal f;raphs In [66] Hayward had established a structural relationship between chordal graphs and weakly chordal graphs by defining a construction scheme on weakly chordal graphs. He noted that the class of chordal graphs can be generated by repeatedly adding a vertex which is not the middle vertex of P3 and showed that weakly chordal graphs can likewise be generated by starting with a set of vertices and no edges and repeatedly adding an edge which is not the middle edge oi P^. The following Theorem states Hayward [66] result in more appropriate way. Theorem 2.19. A graph is weakly chordal graph iff it can be generated in the following manner (i). Start with a graph Go with no edges. (ii). Repeatedly add an edge e-^ to Gy\ to create the Gj such that e^ is not the middle edge of any Pi, of Gj. We give an algorithm using above result for the construction of a sc weakly chordal graph with a given degree sequence {d^ >d-, >...>d„) of sc graph with n vertices. The algorithm-2.6 work as follows: it starts with a graph Gj with no edges on n vertices. Then step-2 repeatedly adds an edge e, between any two vertices to obtain the graph Gj+i such that the added edge is not the middle edge of any P4. Repeating this process if the algorithm obtains the degree sequence as given in input of the algorithm, then it stops and produces a sc weakly chordal graph. The following algorithm constructs sc weakly chuiual giapli willi the given degree sequence on n vertices. Chapter J On sc chorda! and sc weakly chordal graphs Aigorithm-2.6: An Algorithm for the construction of sc weakly chordal graph. Input: A degree sequence {d^ >d-^>... > d„) which has at least one sc graph. Output: A sc weakly chordal graph with degree sequence (t/, >dj>...>d„). Step-1: Start with a graph G\ on « vertices and no edges. Step-2: Add an edge e\ between any two vertices such that added edge e\ is not middle edge of any P4 of Gj. Step-3: Repeat the process of step-2, until to get a graph with degree sequence {d,>d,>...>d„). End. We illustrate algorithm-2.6 by constructing a sc weakly chordal graph on 9 vertices. Suppose we have a degree sequence (6,6,4,4,4,4,4,2,2) and we want to obtain a sc weakly chordal graph of this degree sequence. Algorithm-2.6 starts with 9 vertices as {V|,V2,V3,V4,V5,V6,V7,V8,V9} and no edges as shown by graph G\ in figure-2.9(a). We can add edges on this graph G\ one by one such that added edge is not the middle edge of any Pi, as the rule of step-2 of algorithm-2.6. The overall procedure of the construction of sc weakly chordal graph with the given degree sequence (6,6,4,4,4,4,4,2,2) is obtained as given below and illustrated in figure-2.9. (i). The graph G^ as shown in rigure-2.9(b) can be easily obtained by adding edges ei,e2,eT„en,e<, and ^6 between the vertices {V|,V7}, {v|,V3}, {vi,v4}, {v|,V5}, 52 Chapter 2 On sc chordol and sc weakly chordal graphs {vi,V(,} and {V|,vjj} respectively. From the figure-2.9(b) it is clear that none of the added edges are the middle edges of any P4. (ii). In the graph G3 as shown in figure-2.9(c), added edges are ej, ^8, eg, e\o and e]i. Edge e?, eg and eg are not the middle edges of P4 since they are part of the triangle among the vertices (v), vj, V3}, {v,, V2, V4} and {v|, V2, vt} respectively. Edges eio and eu are obviously not the middle edges of any P4 since they have vertex V7 and vg of degree 1 respectively. (iii). In the graph G4 as shown in figure-2.9(d), the added edges are eu and e^. Edges ei2 and 613 are not the middle edges of P4 since they are part of triangle among the vertices {v|, V3, vj} and {vj, V3, V9} respectively, (iv). The next graph G5 as shown in figure-2.9(e) has added edges as ^,4 and ei5. Both edges are the part of triangle among the vertices {vi, V4, V5} and {vj, V4, V7) respectively. (v). In the graph G(, as shown in figure-2.9(0 the only added edge is e^f,. Edge ei6 is not the middle edge of any P4 since it is part of the chordless cycle of length 4 between the vertices vi, ^^3, V5, and v-,. (vi). The next graph Gj as shown in figure-2.9(g) has added edges as e^ and (?i8- Both edges are the part of triangle among the vertices {v^, V6, V7} and {v], V6, vg} respectively. The algorithm-2.6 stops here, since graph G^ has degree sequence (6,6,4,4,4,4,4,2,2) which is the same degree sequence as given in input of the algorithm. Since there is no violation at any stage of the algorithm, so the 53 Chapter 2 On sc chordal and sc weakly chorda! graphs constructed graph G^ is sc weakly chordal graph with the degree sequence (6,6,4,4,4,4,4,2,2) on 9 vertices. , ^4 .V6 G, • V7 V3 V9 (a) (b) (c) (d) (e) (0 (g) Figurc-2.9 54 CHAPTER THREE Chapter 3 On the recoifnition of same classes of sc perfectly orderable graphs Chapter-3 On the recosnition of some classes of sc perfectly orderable graphs /••':: . \ 3.1 Introduction \ ,/ ^\ - . . ./ In this chapter, we extend our study to the other subclasses of sCperfect graphs, such as sc brittle graphs, sc quasi-chordal graphs and Ps-free sc weakly chordal graphs. One of the important relation between all these 3 classes is that they are subclasses of sc perfectly orderable graphs. In section 3.2, we study brittle and sc brittle graphs and propose a recognition algorithm for sc brittle graphs and then obtain the catalogue of sc brittle graphs. In section 3.3, we study quasi-chordal and sc quasi-chordal graphs and present an algorithm for the recognition of sc quasi-chordal graphs followed by its catalogue compilation. Last section 3.4 is devoted for the same problem forPj-free sc weakly chordal graphs. 3.2 Sc brittle graphs and its recognition Let us first recall an induced P^ with vertices a, b, c, d and edges ab, be, cd then we refer to the vertices a and d as the endpoints, and the vertices b and c as the midpoints of the P^. Based on this notation, Chvatal [25] defined brittle graph as follows: a graph G is brittle if each induced subgraph //of G contains a vertex that is not a midpoint of any P4 or not an endpoint of any P4. 55 Chapter 3 On ihe recoffnilion of some clasaes of sc perfectly orderable graphs In [74], Hoang and Khouzam studied brittle graphs and showed among other things that brittle graphs can be recognized in 0{rr'm) time. Using the concept of brittle order Hoang and Reed [72] gave an algorithm which recognizes brittle graph in Q{n) time. After that Schaffer [120] dealt specifically with the recognition problem for brittle graphs and gave 0(w ) time recognition algorithm derived from definition but this algorithm is much complicated. In a technical report [126], Spinrad and Johnson also designed an O(n^log^n) algorithm for brittle graphs. Later Eschen et al. [42] presented two algorithms for recognition of brittle graphs by direct application of the definition. First they presented an algorithm that requires 0(A7) adjacency matrix multiplication as its bottleneck step which yields an 0(rt^""'') time bound algorithm. Then they presented an algorithm that uses modular (or substitution) decomposition. When used together with known algorithms for modular decomposition and on-line mamtenance of spanning trees, this approach yields an 0{n log ri) time deterministic or 0(^7 ) time randomized recognition algorithm for brittle graphs. In this section, we study the problem of recognition of sc brittle graphs, the proposed recognition algorithm employs the idea similar to those used in above mentioned algorithms. Sc brittle graphs can be recognized using the vertex elimination scheme i.e. if all its vertices eliminated by successive deletion of no-mid and no-end vertices, where a vertex of a graph G is called no-mid if it is not the midpoint of any P4 in G. Similarly, a vertex of a graph G 56 Chapter 3 O" the recognition of some classes of sc perfectly orderoble praphs IS called no-end if it is not the endpoint of any ^4 in G. If a graph G does not contain any induced P4 then all the vertices should be treated as no-mid as well as no-end vertices. No-mid and no-end vertices in sc brittle graphs have same structure. To ensure this we give following Theorem, which relates no-mid vertex to no-end vertex in a sc brittle graph. Theorem 3.1. Let G be a sc graph, then if there exists any no-mid vertex in G, then there also exists a no-end vertex in G. Converse is also true. Proof. Let G be a sc graph, suppose a vertex v be a no-mid in G, then by definition of no-mid vertex it is not the middle vertex of any P4 in G. Now in the complement of the graph G, the same vertex v becomes no-end vertex because the complement of a ^4 is also a P^. Since G is sc graph, therefore if there exists any no-mid vertex in G then there also exists a no-end vertex in G. The same argument is also true for the converse. Hence the Theorem, D The following corollary is immediate from the above Theorem-3.1. Corollary 3.2. Let G be a sc graph, if there exists no no-mid vertex in G then there will be no no-end vertex in G. Converse is also true. To recognize sc brittle graphs, we have to find first a vertex which is either a no-mid or no-end, for this we give an algorithm-3.1, which finds a no- mid or no-end vertex in a given sc graph, moreover it is used as a subroutine in the next algorithm-3.2. Algorithm-3.1, first computes all the induced P^s of the given graph G, then it checks whether the given sc graph G contains any no-mid or no-end vertex or none of these two. 57 Chapter 3 On the recoanilion of some classes of sc perfectly orderable s;raphs Algorithm 3.1: An Algorithm for no-mid and no-end vertices. Input: A sc graph G. Output: Vertex set of no-mid and no-end vertices. Step-1: no-mid set = ^, no-end set = (f) and R^= (j), (where R^ is the set of vertices which are neither no-mid nor no-end in G ) list all the induced A's of G. Step-2: If no induced P^ found then return "Input graph has no induced P^\ Stop. Step-3: select arbitrary vertex 'u' if it is not a middle vertex of any P4 in G, then put the vertex 'w' in no-mid set. else if it is not a end-vertex of any P4 in G, then put the vertex '?/' in no-mid set. else put the vertex 'w' in R,. Step-4: If all the vertices scanned then Stop. Else goto step-3. End. The time complexity of the algorithm-3.1 is as follows. Complexity. Since all the P^'s are generated in 0{n'tn) time [120] and their can be at most 0{n-m) P^'s in G, so step-1 can be done 0{n^in) time. From step-3 to step-4, we require one more iteration of size n, each of which requires 0(/7 m) time. So overall time complexity is Q{rr'm). 58 Chapter 3 On the recofntilion of some classes of sc perfectly orderable graphs Once we compute no-mid vertex set and no-end vertex set, we are in a position to discuss algorithm for recognition of sc brittle graphs. Algorithm- 3.2, which recognizes whether a given sc graph is sc brittle or not works as follows: first it computes no-mid and no-end vertex set in step-1 by using algorithm-3.1, as well as if it finds no induced P^s then algorithm decides that the given graph is sc brittle. Now if both the sets i.e. no-mid and no-end vertex sets are empty then algorithm terminates and gives output that the given sc graph is not sc brittle, otherwise algorithm proceeds to the next step i.e. step-4. In this step, it deletes vertex either from no-mid or no-end vertex set (note that the choice of no-mid or no-end vertex to delete is arbitrary since the deletion of a vertex cannot cause another vertex to become the midpoint or endpoint of a ^'4). Then the algorithm repeats the procedure from step-1 to step-4. In this way if all the vertices are eliminated then algorithm-3.2 decides that the given sc graph is sc brittle. Algorithm-3.2 is as follows. Algorithm 3.2: An Algorithm for recognition ofsc brittle graph. Input: A sc graph G. Output: "G is sc brittle graph" or "G is not sc brittle graph". Step-1: Compute the no-mid and no-end vertex set (using algorithm-3.1), and elimination order = ^ Step-2: If graph has no induced A's then print "Graph G is sc brittle". Stop. Step-3: If no-mid set = ^ = no-end set, then 59 Chapier 3 0» I he recof^nilion of some classes of sc perfectly orderable eraphs return "G is not sc brittle". Stop. Step-4: eliminate vertex 'w' either from no-mid set or no-end set. Put 'u' in elimination order. Step-5: update the graph and goto step-1. Step-6: If all the vertices of G are eliminated in this way, then "G is sc brittle" and print the vertices of the elimination order, else "G is not sc brittle". End. The time complexity of the algorithm-3.2, can be computed as follows. Complexity. In algorithm-3.2, the bottleneck is step-1, which requires 0(A7 m) time. All the other steps have lower complexity. So the overall time complexity is 0{n^m). The following result justifies the claim of algorithm-3.2. Th eorem 3.3. Algorithm-3.2 checks whether an input sc graphs is sc brittle or not correctly. Proof. By the definition of brittle graphs, its each induced subgraph contains either a no-mid or no-end vertex. So we start with a sc graph G and look for no- mid or no-end vertex, if found, remove that vertex (let it be u). Now by definition of brittle graphs, G - u again contains no-mid or no-end vertex, if 60 Chapter 3 On ihe recoenilion of some classes of sc perfectly orderable eraphs found, remove that vertex again. Clearly if all the vertices are removed in this manner, then we are left with no vertex and the graph G is sc brittle. If while deleting the vertices we found that there does not exist any no-mid or no-end vertex in any subgraph of G, then at that stage we decide that the graph is not sc brittle. Hence the Theorem, n To illustrate algorithm-3.2, we consider the following sc graphs G] and G2as shown in rigure-3.1(a) and figure-3.1(b) respectively. V7 Ve ^VcT v? J^ Vl, (V« (a) (b) Figurc-3.1 Let graph G] be the input to algorithm-3.2. Step-1 of algorithm-3.2 computes its no-mid set as {v3,v4} and no-end set as {vj,v^}, by calculating all induced P4'S as [V|,V5,V2,V6], [V|,V8,V6,V2], [V|,Vs,Vf„V3], [V2,Vi,V^,V4], [V2,V5,Vij,V4], [V2,V6,V8,V4], [V2,V7,V1,V4], [v2,Vl,V8,V4], [Vi,V(„V2,Vi], [Vi,V(„V^,V4], [V3,V6,V8,V5], [V3,V7,V,,V4], [V'3,l/7,V,,V5], [V3,V7,V2,V5], [Vi,V^,V^,V4], [V3,^7,V8,V5], [V4,V|,V7,Ve] and b'5,v^,v^,V(,] using algorithm 3.1. Since both the no mid and no-end sets are not empty thus algorithm proceeds to step-4. In step-4 any vertex from either no- Chaplcr i On the recoeniiion of some classes of sc perfectly orderable graphs mid or no-end vertex set can be deleted, let it be V3. The graph is updated in step-5 and then algorithm repeats the procedure. In this way algorithm-3.2 successfully eliminates all the vertices of G| and produces output as G\ is a sc brittle graph. The overall procedure of elimination of vertices from no-mid and no-end set can be seen in figure-3.2, where a vertex which is enclosed by dotted line is deleted vertex. VT ^6 JVcT V2 ^-^ =>^^ Vl, /vs (a) '. Vj / V V6 < 1 "^^ ^''< 1 "' 1^5 < ^ ^2 (h) V2 (0 / (c) r-vg (d) (g) Figure-3.2 62 Chapter 3_ On (he recognition of some classes of sc perfectly orderable eraphs Now let the graph d as shown in rigure-3.1(b) be the input to algorithm-3.2. Step-1 of algorithm-3.2 computes the no-mid and no-end vertex set and finds no-mid set = = no-end set and R^ = {vi,V2,V3,V4,V5,V6,V7,vg} by using aigorithm-3.i. All the induced /'4's of G2 are [vi,V6,V2,V5], [vi,V7,V2,V5], [Vl,V7,V,,V5], [V2,Vs,V|,V4], [v^,Vi,V2,Vf,], [V3,V7,y,,V6], [V3,V7,V2,V6], [V3,V7,V4,V6], [V3,V8,V,,V4], [V3,Vs,V,,V6], [V3,V8,V2,V6], [V4,V,,Vg,V5], [V4,V6,V2,V5], [V4,V6,V2,V8], [V4,V7,V2,V5], [V4,V7,V'2,V8], [V4,V7,V3,V5], [V4,V7,V3,V8], [V5,Vg,Vi,V6] and [V5,V8,V|,V7]. Since both the sets i.e. no-mid and no-end are empty so algorithm-3.2 declares that the input graph Gj is not sc brittle graph. 3.2.1 Catalogue compilation of sc brittle graphs Using algorithm 3.2, we compile the catalogue of sc brittle graphs with at most 17 vertices from the available catalogue of sc graphs with at most 17 vertices. Number of vertices n 4 5 8 9 12 13 16 17 Nitmher ofsc graphs 1 2 10 36 720 5600 703760 11220000 Number ofsc briale graphs 1 1 6 6 82 82 5912 5912 Tablc-3.1 63 Chapter 3 On the recognilion of some classes of sc perfectly orderable graphs 3.3 Quasi-chordal and sc quasi-chordal graphs Quasi-chordal graphs were introduced by Voloshin [147] as a generalization of cliordal graphs. Hoang and Mahadev also gave the similar concept in [75], where they called quasi-chordal graphs as good graphs. The following Theorem was known to researchers and referred in the literature but its proof had never been published. Recently Gorges et al. [57] proved the Theorem. Theorem 3.5 [57]. For a graph G, the following conditions are equivalent: (i) G is quasi-chordal. (ii) G does not contain a latticed subgraph as an induced subgraph. (iii) G admits a good order. where an order i'i Recently, in [79] Hoang et al. reported the following result. Theorem 3.6 [791. If C is a weakly chordal graph such that every pair of squares meet in a non-edge, then G is a quasi-chordal graph. The above Theorem does not guarantee that if at least one pair of squares meets in an edge then whether G is quasi-chordal or not? To answer this question, we further study sc weakly chordal graphs for above case and obtain the following result, i.e. when at least one pair of squares meets in an edge. 64 Chaplcr 3 On the recof Theorem 3.7. Let G be a sc graph, such that at least one pair of squares meets in an edge then G contains one or more of the following graphs as induced subgraphs. U (1 U m w - # D, ,^\ >. K, D, ,S-.D, , Proof, Let G be sc graph, suppose there exists at least one pair of squares which meets in an edge. Now whenever two squares meet in an edge then the graph formed in this way will always have six vertices only, however they may have seven vertices but in this case the graph has no edge common as shown below in figure-S.S, so we do not consider this graph , Figurc-3.3 Now consider a six cycle graph with vertices Vi,v2, vj, V4, V5,V6 with edges V1V2. V2V3, V3V4, V4V5 V5V6, v^V]. We add an edge between the vertices vj and V5 as V2V5 This obtained graph is shown below in figure-3.4 as graph D\. The graph D\ has clearly two squares as {v\,V2,Vi,Vh], {V2,V3,V4,V5} and they meet on edge y2V5- 65 Chapter 3 On the rccof>nilion ot some classes of sc perfectly orderable graphs Vl D, V6 V, V4 Figure-3.4 If we add edges on graph D| one by one between the non-adjacent pair of vertices then we get following graphs D2, D3, D^, D5, D^, D-i, Dg and D9 as follows; The graphs Dj and Dj as shown below in rigure-3.5 are obtained by just adding a single edge on graph D\ between the vertices vi,v4 and V|,V3 respectively. (a) (b) Figurc-3.5 The graphs D4, D5 and Df, as shown below in flgure-3.6 are obtained by adding edges on D\ as (vi,V4), (v3,V6) for D4, edges (vi,v4), (v|,V3) for D5 and edges (Vl,V3), (V4,V6)forZ)6. k'l V 2 v., >. '< V5 Vj (a) (b) Figurc-3,6 Similarly graphs D7, Dg and D9 as shown in figure-B.V, can be obtained from D\ by adding the edges as follows; (v|,V4), (vi,v3), (vi,V(,) for D7, (vi,V4), (vi,V3) {V4,V(,) for Ds and (vuv^), (vi.vj), (v3,V6), ivA,V(,) for D9. (a) 66 Chapler 3 On the recoenilion of some classes of sc perfectly orderable sraphs Clearly graphs D| to Dq are the only possible graphs on six vertices such that two squares meet in an edge. Hence the Theorem, D We note that the graph D3 given in figure-3.5(b) contains an induced cycle C5: v\, vj, V4, V5, V(, and V] and the graph D^ is the complement of induced cycle Ce as given in figure-3.6(c). This observation shows that both the graphs £>3 and Ds are not weakly chordal, so we have the following result. Corollary 3.8. Let C be a sc weakly chordal graph such that at least one pair of squares meet in an edge then G does not contain the following graphs as induced subgraphs. 3.3.1 Recognition of sc quasi-chordal graphs Many graph classes are defined or characterized in terms of an elimination scheme. For example chordal graphs are defined as having no induced cycles of length greater than 3. They are characterized as those graphs which have an elimination scheme with the property that neighbors of Vj induce a clique. Trees can be characterized as those graphs such that every eliminated vertex (except for v„) has degree I in the remaining graph. Similarly for quasi- chordal graph there also exists an elimination scheme, which is ensured by the following result given by Voloshin [146]. 67 Cliaplcr 3 On the recof;nilion of some clusscs of sc perfectly ordcrable eraphs Theorem 3.9 11461. Let G be a graph on n vertices. Then G is quasi-chordal if and only if each of its induced subgraph contains a simplicial or co-simplicial vertex. Based on the above result, recognition of quasi-chordal was first studied by Voloshin in [146] and he proposed an 0{rf) time algorithm. Later Spinrad [128] proposed an 0(/7 ) time algorithm. Hoang [78] also independently proposed an 0{nm) time algorithm. Much recently Gorgos et al. [57] also proposed an 0{nm) time algorithm for recognizing quasi-chordal graphs. Quasi-chordal graph may admit many different elimination schemes i.e. if it has eliminated scheme only on the basis of simplicial vertices then the graph is chordal and if the eliminated vertices are only co-simplicial then the graph is co-chordal. Now if the vertices of quasi-chordal graphs are eliminated by first removing simplicial and then co-simplicial vertices then the graph is called semi-chordal. The following lemma shows the connection between the simplicial vertices and no-mid vertices as well as co-simplicial vertices and no- end vertices. Lemma 3.10. Every simplicial vertex is no-mid vertex and every co-simplicial vertex is no-end vertex. Converse need not to be true. Proof. Since simplicial vertex cannot be middle vertex of any induced P^ and every induced P4 always contains two, P->, as a induced subgraphs. This implies that the middle vertices of both ^3 are always mid vertices of A, therefore simplicial vertices cannot be middle vertex of any Pn. Hence they are always 68 Chapter 3 On the recoKnilion of some classes nf sc perfectly orderable graphs no-mid vertex. Now suppose x is any end vertex of an induced P4 then the non- neighbors of X will never form stable set as one of its non-neighbors is an end- vertex and the other is mid-vertex, therefore end vertices of any P4 cannot be co-simplicial. Hence co-simplicial vertex is always no-end vertex. Now for converse consider a chordless cycle of length 4 (i.e. C4), then each vertex of this chordless C4 is no-mid but not simplicial. Similarly in the complement of this C4(i.e. 2K2), every vertex is no-end but not co-simplicial. Hence the result, D The following result shows the relation between simplicial and co- simplicial vertices in sc graphs. Theorem 3.11. Let G be a sc graph, if there exists any simplicial vertex in G then there also exists a co-simplicial vertex, converse is also true. Proof. Let G" be a sc graph, suppose a vertex v be a simplicial in G, then by definition of simplicial vertex, all the neighborhood vertices of i'are adjacent to each other. Now, in the complement of the G. same vertex v and adjacent vertices make an independent set of vertices i.e. v becomes co-simplicial \nG. Since G and G are isomorphic, thus in G, there also exist a co-simplicial vertex. The same argument also holds for the converse, n The following corollary is immediate from the above Theorem. Corollary 3.12. Let G be a sc graph, if there exist no simplicial vertex in G then there exist no co-simplicial vertex, converse is also true. 69 Chapter 3 On ihe recoenilion of some classes of sc perfectly orderable graphs For recognizing whether a sc graph is quasi-chordal graph, we use a different method as compare to algorithms discussed in [128], [57]. The basic difference between the algorithms in [128], [57] and the one given here is that we find the simplicial and co-simplicial vertices within the set of no-mid and no-end vertices while the other algorithms find simplicial and co-simplicial vertices in whole graph. So to decide whether a vertex is simplicial or co- simplicial or not first we present an algonthm-3.3 for simplicial and co- simplicial vertices. Further algorithm-3.3 is used as subroutine in algorithm- 3.4. The algorithm-3.3 is as follows. Algorithm 3.3: An Algorithm for recognizing simplicial and co-simplicial vertices. Input: A graph G and a vertex 'z/'. Output: Return either simplicial or co-simplicial vertex. Step-1: Compute neighborhood of'w' i.e. N{u). Step-2: If yV(M)induces a complete subgraph of G, then return " 'w' is a simplicial vertex" Stop. else compute its non-neighbors N'{u)'\.t. N'{u)= V{G)-N[u] If A'Yw) induces a stable set of G, then return " 'w' is co-simplicial vertex" Stop, else return " 'z/' is neither simplicial nor co-simplicial vertex". End. Complexity. The time complexity of the algorithm-3.3 is 0(A7 ) by [79]. 70 Chapter 3 On I he recoertilion of some classes of sc perfectly orderable eraphs The algorithm-3.4 as given here mainly depends on finding no-mid and no-end vertex set in the input graph G. As soon as it computes no-mid and no- end vertex set, algorithm goes for the search of simplicial and co-simplicial vertices within the set of no-mid and no-end vertices. If no-mid and no-end vertex sets do not contain any simplicial and co-simplicial vertices respectively then at the initial stage of algorithm it is possible to get the output i.e. the input graph is not quasi-chordal. Algorithm 3.4: An Algorithm for recognizing sc quasi-chordal graph. Input: A sc graph G. Output: "G is sc quasi-chordal graph" or "G is not sc quasi-chordal graph". Step-1: Compute set of no-mid, no-end,/?,. (using algorithm-3.1). and elimination order = (p Step-2: If all the vertices of no-mid and no-end set are simplicial and co-simplicial respectively ( using algonthm-3.3). and no-mid u no-end = vertex set of graph, then print " G is sc quasi-chordal graph" and print "vertex set of G" Stop, else if no-mid and no-end set contains no simplicial and no co-simplicial vertices respectively, then print "G is not sc quasi-chordal graph" Stop. Step-3: select either a simplicial vertex from no-mid set or co-simplicial vertex from no-end set (let this vertex be u) remove vertex 'w' and put'«' in elimination order. Chapter 3 On ihe recognition of some classes of sc perfectly orderable graphs Step-4: update the graph and goto step-1. Step-5: If all the vertices are eliminated in this way then print "C is sc quasi-chordal graph" and print "elimination order set" Else print "G is not sc quasi-chordal graph" End. The time complexity of the algorithm-3.4 is as follows. Complexity. Algorithm-3.4 first uses algorithm 3.1 in step-1, which has time complexity 0{n'm). All the other steps require lesser time. Hence the overall time complexity is 0{n in). The correctness of algorithm-3.4 is as follows. Theorem 3.13. Algorithm-3.4 checks whether an input sc graphs is quasi- chordal or not correctly. Proof. Simplicial vertices are no-mid vertices and co-simplicial vertices are no-end vertices. So while running the algorithm-3.4, when we compute set of no-mid and no-end vertices simplicial vertex always lies in no-mid and co- simplicial vertex lies in no-end set. Now from the statements (iii) and (i) of Theorem-3.5, it is clear that if we eliminate all the vertices in such a way then the resulting graph is always a quasi-chordal. Hence the Theorem, D To illustrate algorithm 3.4 we consider the sc graphs Ci and Gi on 9 vertices as shown in figure-3.8(a) and figure-3.8(b) respectively. Let the sc graph G\ as shown in figure-3.8(a) be input to the algorithm-3.4. Step-1 finds no-mid set, no-end set and R^, as follows. no-mid = {vi,v^}, no-end = {v7,vg}, R^ = {v\,V2,Vi,V(„v^}, 11 Chapter 3 On the recosniiion of some classes of sc perUcltv orderable graphs Figurc-3.8 this is done by using algorithm-3.1. Now step-2 of algorithm-3.4 finds that vertices V3, V4 ft-om no-mid set and vertices V7, vg from no-end set. These are simplicial and co-simplicial vertices respectively (this is done by using algorithm-3.3). Although all the vertices of no-mid and no-end sets are simplicial and co-simplicial respectively, but no-miduno-end * vertex set of graphs i.e. {v3,V4}u {v7,V8} * {v\,V2,Vf„v^,Vi,Vf„VT,v^v<)} therefore algorithm-3.4 proceeds to step-3. In this step let vertex V3 be eliminated first. Then step-4 updates the graph and repeats the process again. Eventually algorithm-3.4 successfully eliminates all the vertices of G\ one by one and produces the eliminaUon order set = {vT„v4,v-,,vi,vg,v6,v5,V2,v9}. Thus algorithm-3.4 decides that the input graph G\ is sc quasi-chordal graph. The overall procedure of elimination of vertices can be seen in figure-3.9. Let graph Gj as shown in figure-3.8(b) be the input to algorithm-3.4. The algorithm first finds its no-mid set, no-end set and R^ as follows, no-mid = {V|,V4,V5,V6}, no-end = {v2,V3,V7,vs} and Rv = {V9}, 73 Chapter 3 On the recofinilion of some classes of sc perfectly orderabte graphs this is done by using algorithm-3.1. In step-2 the algorithm-3.4 does not find any simpHcial vertex in no-mid set and no co-simplicial vertex in no-end set, therefore the algorithm terminates and decides that the input graph G2 is not sc quasi-chordal graph. Figurc-3.9 74 Chapter 3 On ihe recof-niiion of some classes of sc perfectly orderable graphs 3.3.2 Catalogue compilation of sc quasi-chordal graphs Using algorithm 3.4 we compile the catalogue of sc quasi-chordal graphs with at most 17 vertices from the available catalogue of sc graphs with at most 17 vertices. Number of vertices n 4 5 8 9 12 13 16 17 Number of sc graphs 1 2 10 36 720 5600 703760 11220000 Number ofsc quasi-chordal graphs I 1 5 5 62 62 2406 2406 Table-3.2 3.4 A-free weakly chordal and A-free sc weakly chordal graphs Weakly chordal graphs are not necessarily perfectly orderable [69], however many known classes of perfectly orderable graphs are weakly chordal (such as chordal, quasi-chordal and brittle). In [77], Hoang showed that determining whether a graph is perfectly orderable remains NP-complete for the class of weakly chordal graphs. In 1990 Chvatal conjectured [23] that every Ps-free weakly chordal graph is perfectly orderable. Hayward [69] proved this conjecture by presenting a polynomial time algorithm to find a perfect order of any Pj-free weakly chordal graphs. Theorem 3.15 [69|. /'j-free weakly chorda! graphs are perfectly orderable. On the other hand, it is known that P5 -free weakly chordal graphs are not necessarily perfectly orderable [69]. In (his section we show that Pi-fveQ sc 75 Chapter 3 On the rccoi;nilion of some cUisxe.s ol sc nerfeclh orderable firaphs weakly chordal graphs are perfectly orderable. To prove this we need the following result. Theorem 3.16. Let G be a sc graph. Then G is fs-free if and only if it is Ps- free. Proof. Let G be a sc graph. Suppose G contains P^ with vertices vi, vj, V3, V4 and vj. Then these vertices v,, vj, v,, V4, V5 induces a house {Pi)\n G as shown below in figure-3.10 V3 V5 ^1 V2 V, V4 V5 • • • • • V2 Figurc-3.10 Since G is sc, so if G contains P5 it also contains fs, therefore if G is F^-free then G is Ps-free. Conversely, suppose G contain Ps with vertices ^1,^2,^35^4,^5, then in G these vertices become a chordless path, P5 as shown above in figure-S.lO. Hence G is Ps-free implies G is Ps-free. Hence the Theorem, a Using above Theorem, we get the following result. Theorem 3.17. P5 -free sc weakly chordal graphs are perfectly orderable. Proof. By Theorem-3.16, it is clear that in the case of sc weakly chordal graphs Ps-free sc weakly chordal graph class and Ps-free sc weakly chordal graph class are exactly same i.e. whenever sc weakly chordal graph is Ps-free then it is also Ps-free, and vice-versa. So by Theorem-3.15 and Theorem-3.16, 76 , ,, A 73,' If^ Chapter 3 On ihc recognition of some classes of sc perfeci/v orderablei^j^hi I r \ - V Ps-free sc weakly chordal graphs are perfectly orderaBfe^^HericeV the Theorem, n While studying Pj-free sc weakly chordal graphs another graph class known as charming graph [76] has almost same structure as of Ps-free sc weakly chordal graph. A vertex v in a graph G is said to be charming if v is not end vertex in a P5 in G, is not end vertex in a P5 in G and does not lie on a C5 in G. A graph is charming in which every induced subgraph has a charming vertex [76]. In the same paper Hoang proved that every charming graph is perfectly orderable, moreover he showed the following. Theorem 3.18 [76|. Every weakly chordal graph with no induced P5 and Pb is charming. Now for sc weakly chordal graphs we have following Corollary. Corollary 3.19. Every sc weakly chordal graph with no ^5 and ?& is Charming. Proof. Since sc weakly chordal graph is isomorphic to its complement. Therefore the condition on Pb in the Theorem-3.18 is reduced to P^ for sc weakly chordal graph. Hence the result, D We know that if G is Pj-free then G is /'6-free. Thus we have the following result. Corollary 3.20. Every P^ free sc v.'cakly chordal graph is Charming. Proof. Follows from Corollary-3.19. D 77 Chapter 3 On the recof;nilion of some classes of sc perfecllv orderable graphs Now it is clear from Coroliary-3.20 that if we recognize Ps-free sc weakly chordal graphs then these graphs are also charming. 3.4.1 Recognition of Fs-free sc weakly chordal graph In [69] Hayward proposed an 0{rf') time algorithm for recognizing a Ps-free weakly chordal graphs. Infact he used complex concepts like separating sets and handles for their algorithm, which is not easy to implement. Recently Nikolopous and Palious [96] also presented parallel algorithms for recognizing Ps-free and Ps-free weakly chordal graphs. Their algorithms are based on one of the following result. Theorem 3.21 [96|. Let G be a weakly chordal graph with vertex set V{G). Then G is Ps-free weakly chordal if and only if none of its edges is a Ps- witness. An edge e = {a,b) is Ps-witness in a weakly chordal graph if there exist a vertex u e V{G) - N{e\ such that N{u) contains vertices from both A{a,e) and A{h,el (where A{a,e) - N{a) - N{b], A{b,e) = N{b) - N[a] and N[e] = N[a] fl Nib]). We use method similar to [96] for recognizing Ps-free sc weakly chordal graphs. Since Ps-free sc weakly chordal graph class and Ps-free sc weakly chordal graph class are same, so we can use above result for recognizing Ps- free sc weakly chordal graphs. The foHcvving algorithm recognizes Pj-free sc weakly chordal graphs as well as sc charming graphs. 78 Chapter 3 On Ihe recoeniiion of some classes of sc perfectly orderahle graphs Algorithm 3.5: An Algorithm for recognizing Ps-free sc weakly chordal graph. Input: A sc weakly chordal graph G. Output: G is Ps-free sc weakly chordal graph or G is not /'s-free sc weakly chordal graph. Step-1: Select any arbitrary unmarked edge, say e = {a, b) If no unmarked edge remains Print "Ps-free sc weakly chordal graph" Stop. Step-2: Find yV(o), N{h\ N{a] and N{b] Step-3: Compute A{a,e) = N{a) - N{bl k{b,e) = N{b) ~ N[a\, N{e] = N{a] - N{b\ and R^ = V{G) - N[e] Step-4: for every vertex w 6/?v if N{u) contains vertices from both A{a,e) & A{b,e), then Print "G is not Pj-free sc weakly chordal graph". Stop, else mark the edge e. goto step-1 End. Now, we compute the time complexity of the algorithm-3.5. The graph G is assumed to be given in its adjacency list representation. ComplexitY. We compute the complexity of each step separately. In step-1, since we have to analyze each edge as worst case, so it takes 0(m)-time. Step-2 is executed for each vertex of edge e, taking maximum {n-\) executions for each vertex so this step takes 0(A7-1) = 0(/7) time. Step-3 takes obviously 79 Chapter 3 On ihc rccof-nilion of some classes uf sc perfectly orderable graphs constant time. Now the step-4 takes 0{n-\) = 0{n) time to be executed as worst case. Taking into consideration the time complexity of each step of algorithm, we have total complexity 0{mn). Theorem 3.22. Algorithm-3.5 checks whether an input sc weakly chordal graphs is Ps-free sc weakly chordal or not correctly. Proof. The correctness of algorithm follows from Theorem-3.21. D To illustrate algorithm-3.5, we consider the following sc weakly chordal graphs G\ and Gj as shown in figure-3.11(a) and figure-3.11(b) respectively. V4 (a) (b) Figurc-3.11 Let us input graph G\ in algorithm-3.5 as shown in figure-3.11(a). Step-1 selects an edge e = {vsyi) arbitrarily. Step-2 computes A'(v3) = {vi,V(„v-j,Vi}, yV(v7) = {vi,V2,V3,V4}, N\v^] = {v3,V5,V6,v7,vx} and N{vj] = {V|,V2,V3,V4,V7}. Then step-3 computes A{v^„e) ~ N{vi) - N{vj] = {vi,V(„v^], A{vi,e) = N{VT) - N{v^] = {vi,V2,V4}, N{e] = MvjinMv,] = {V3,V7} and R, = V{G) - N{e] = {Vl,V2,V4,V5,V6,V8} 80 Chapler 3 On the recoenilion of some classes of sc perfectly orderable graphs Now step-4 checks for all the vertices of ^v, i-e. let M = vi then iV(vi) = {v4,V7,vx}. Now it is clear that A^(V|) contains a vertex V7 which does not lie either in Aivi,e) or in A{v^,e). Similarly Ni^vj), Niv^}, Nivs), //(vg) and Niv^) contain vertices V7, ^7, V3, V3, V3 respectively which are neither in A{vi,e) nor in A{v^,e). Hence edge e = {vi,vy) is not Ps-witness and edge e = {vi,v^) is now marked edge. Now aIgorithm-3.5 checks for every other edge of G| and finally finds no edge as fs-witness. Hence aIgorithm-3.5 decides that the input sc weakly chordal graph G, is Ps-free sc weakly chordal graph. Now consider the other sc weakly chordal graph Gj as shown in figure- 3.11(b). Step-1 of algorithm-3.5, selects an edge e = (vj.vs) arbitrarily then step- 2 computes Nivj) = {vj v^, V7}, Niv^) = {v,, vj. v^}, N[v2] = {vj, Vy v^, V7} and N[vi] = {v| V2, V5 vs}- Then step-3 computes A(v2,e) = Nivj) - N[vs] = {v(,y^}, Aivi,e) = NiVi) ~ NIV2] = {vi.vs}, N[e] = NIV2] H N[v,] = {v2,v,} and R, = V{G) ~ M^l = {*'i. ^3, ^4. ^'6, V-,, vg}. Now step-4 checks i.e. let w = V] then A^(V]) = {V4 ^5, V-,, vg}. Now it is clear that N{vf) contains vertices V7 & vg which lie in A{v2,e) and A(yi,e) respectively. Hence step-4 decides that the edge e = {v2,vs) is /'5-witness, therefore the algorithm-3.5 decides that the input sc weakly chordal graph G2 is not PyfrsQ sc weakly chordal graph. 3.4,2 Catalogue compilation oi P^-free sc weakly chordal graphs We compile the catalogue of Fj-free sc weakly chordal graphs with at most 17 vertices from the available catalogue of sc graphs with at most 17 Cliaplcr 3 On the recof:nilion of some classes ol sc perfectly orderable ffraphs vertices. Moreover all the graphs which are Ps-free sc weakly chordal graph are also sc channing. Number of vertices n 4 5 8 9 12 B 16 17 Number ofsc graphs 1 2 10 36 720 5600 703760 11220000 Number oj Pyfree sc weakly chordal I 1 4 4 23 23 275 275 graphs Table-3.3 82 CHAPTER FOUR Chapter 4 On ihe oviimizaiion pruhtems of some classes of sc perfect erophs Chapter-4 On the optimization problems of some classes of sc perfect graphs 4.1 Introduction The problem of finding a maximum clique, a minimum colouring, a maximum stable set and a minimum clique cover of a graph is known as optimization problems. These four problems are NP- hard in general [48]. In [59] Grotschel, Lovasz and Schrijver developed a polynomial time algorithm to solve these four optimization problems for perfect graphs. Although they gave polynomial time algorithm for perfect graphs, but the algorithm is extremely difficult; it uses the ellipsoid method from linear programming theory as a subroutine. No purely combinatorial algorithm is known for solving these problems on perfect graphs. This is the reason that the researchers are still studying optimization algorithms for perfect graphs and subclasses of perfect graphs too. In this chapter we also study these optimization problems for sc perfect graphs and their subclasses namely, sc chordal graphs, sc weakly chordal graphs, sc quasi-chordal and sc brittle graphs. The section 4.2 deals with chromatic number of sc perfect graphs. In section 4.3, we discuss optimization 83 Chapter 4 On the oplimizalion problem.'! of some classes of sc perfect graphs problem for sc weakly chordal graph followed by proposed algorithms. We extend our study for the class of sc perfectly orderable graphs in section 4.4. Optimization problems can be divided mainly in two cases, weighted versions and unweighted versions. In this chapter we consider only unweighted problems and their algorithms. 4.2 On the chromatic number of sc perfect graphs In general obtaining the exact value of the chromatic number of a graph is quite difficult. However researchers had obtained bounds for the chromatic number of graph and several classes of graphs [82] and [139]. In this section we also obtain bounds on the chromatic number of sc perfect graphs and using the result given by Sridharan and Balaji [131], we show that the upper bound is attained iff the graph is sc chordal. The following result on the chromatic number of a graph and its complement was obtained by Nordhaus and Gaddum [98] Theorem 4.1|98]. For any graph G with /? vertices the following holds. (i) \2^]<%{G) + z{G) (ii) n The following result gives bounds for the chromatic number of a sc graph. Corollary 4.2. Let G be a sc graph with n vertices. Then (i) [2i^] {[[) \,l4j^\]< Z{G)<2p + \ when « = 4/7 + /. 84 Cliiiplcr 4 On ihe optimizalion problems of some classes of sc perfect graphs Proof, (i) Let G be a sc graph with 4p vertices. We note that /(G) = xiG) since G is sc graph. So by theorem-4.1, 7(G) should satisfy the relation given by (4.1). [274^1 <2;}r(G)<4p + l and Ap<{x(G)f< Ap + \ (4.1) Then it follows that \'2-fp']< j(G) < 2p (ii) Let G be a sc graph with Ap+\ vertices. We note that /(G) = /(G) since G is sc graph. So by theorem-4.1, /(G) should satisfy the relation given by (4.2). r2V4^1<2/(G)<4p + 2 Ap + 2\2 and 4p + I<(/(G))-< .(4.2) Then it follows that \^Ap+\] In [ 131 ] Sridharan and Balaji gave the following result. Theorem 4.3. Let G be a sc graph. Then G is chordal iff co{G) = 2p for n = 4p and co{G) = 2p+\ for n = 4p+\. The following Corollary is immediate from the above theorem. Corollary 4.4. Let G be a sc perfect graph. Then G is chordal if and only if (i) /(G) = 2p when n = 4p (ii) /(G) = 2;?+l when/7 = 4/;+l. Proof. The proof follows from the fact co{G) = /(G) for perfect graphs, D 85 Chapter 4 On the opiimizalion problems of some classes of sc perfect graphs Next result gives the bounds for the chromatic number of a sc perfect graph. The upper bound given by this result is attained for sc chordal graphs. Theorem 4.5. Let G be a sc perfect graph. Then (') I v4p p/(G) < 2jO when/? = 4/7 (ii) I ^4p+\ \< z{G) <2p + \ when n = 4p+L Moreover, the upper bounds given by (i) and (ii) are attained if G is chordal graph. Proof. Follows from Theorem-4.2 and Corollary-4.4. n From the above result it is clear that all other classes of sc perfect graphs such as sc quasi-chordal, sc brittle and sc weakly chordal graphs will have lower values of /(C) from that of sc chordal graphs. 4.3 Contraction method and optimization algorithms Contraction of two vertices into a new vertex is an important tool for designing optimization algorithms for many classes of perfect graphs such as Perfectly contactile graphs, Meyniel graph and various others [108]. In 1982, Fonlupt and Uhry [46] first observed and proved that if G is a perfect graph and {x, y} is an even pair in G, then the graph G^y is also perfect and has the same chromatic number as G. Since then, contraction of an even pairs have become an important tool for proving that certain classes of graphs are perfect and designing their optimization algorithms. However, the problem of deciding if a graph contains an even pair is NP-hard in general graphs [14]. 86 Chapter 4 On the oolimizaiion problems of some classes of sc perfect graphs In [73] Hoang and Maffray proved that every weakly chordal graph contains an even pair. However contracting even pair in a weakly chordal graph does not always yield a new weakly chordal graph, for example adding an edge between the endpoints of /'2k+i (^>3) produces a C2k. Therefore to overcome this problem and develop a polynomial time optimization algorithm for weakly chordal graphs, Hayward et al. in [67] defined a two-pair, a special case of even pair. Since then optimizing algorithms for weakly chordal graphs mainly depend on the method of two-pair contractions. In order to solve the optimization problems for sc weakly chordal graphs, we also follow the method of contraction of two-pair in the case of maximum clique and minimum coloring, while for the solution of maximum stable set and minimum clique cover we propose a different method namely co- pair edge contraction method. 4.3.1 Optimizing algorithms for sc weakly chordal graphs Optimizing algorithms for weakly chordal graphs were first studied by Hayward et al. in [67]. They gave an 0((/7 + m)n') time algorithm for maximum clique and minimum coloring, while algorithms for maximum stable set and minimum clique cover problem take 0(A7'^) time. The maximum stable set and minimum clique cover problem are essentially solved by running the former algorithm on the complement of the graph. Later Spinrad and Sritharan [127] also studied the same problems but for only weighted versions. Recently, Hayward el al. [68] proposed an 0{nm) time algorithm, using the concept of 87 Chanter 4 On the opiimizaiion problems of some classes of sc perfect graphs handles for weakly chordal graphs. However, they improved the time complexity of the algorithm but because of the involvement of handles it becomes much complicated and not easy to implement. For finding the maximum clique and minimum coloring for sc weakly chordal graphs, we present an 0{mr?) time algorithm, which is based on contraction of two-pair i.e. find a two-pair and modify the graph by adding an edge between them. We now describe algorithm-4.1, which computes the maximum clique and minimum coloring for sc weakly chordal graphs. Algorithm-4.1, takes sc weakly chordal graph G as input. Step-1 of algorithm-4.1 finds a two-pair {x, y) of G, using algorithm-2.1. Step-2 Computes N{x)vjN(y) = W{\Qi). In step-3 we replace vertices x and y with a new vertex z. Step-4 adds edges zw such that weW \x\ this way the graphs that come later in the sequence have strictly less vertices than the graphs those came earlier in the sequence. Step-5 repeats the process until no non-adjacent vertices are left in the graph. Eventually in the last graph all the vertices are mutually adjacent to each other, whose order is equal to the clique number and chromatic number of the input graph G. The algorithm 4.1 is as follows. 88 Chaplcr4 On the oplimization problems ol some classes of sc perfect craphs Algorithm-4.1: An algorithm for finding maximum clique and minimum coloring for sc weakly chordal graph. Input: A sc weakly chordal graph G. Output: (o{G) and;}f(G) for sc weakly chordal graph G. Step-1: Find a two-pair {x,y} of G. (using algorithm-2.1) Step-2: Compute N{x)uN(y) = ^(let) Step-3: Replace(xv -> z), where z is new vertex. Step-4: Add edges ZH'such that we PF. Step-5: Repeat the process from step-1 to step-4, until no non-adjacent pair of. vertices are left. Step-6: Order of last graph = co{G) = /(G). End. Complexity. Since a two-pair can be found in 0{mn) time using algorithm- 2.1. From step-2 to step-4 can be done in linear time. Now step-5 repeats the process n times, so overall time complexity of algorithm-4.1 is 0{mn ) The correctness of algorithm-4.1 rely on the following fact, which are due to Meyneil [93]. Theorem 4.6. Let x and y be two vertices of a graph G that are not joined by a chordless path with 3 edges. Then (o{G{xy -> z)) = a){G). Theorem 4.7. Algorithm-4.1 finds a maximum clique and a minimum coloring ofG. 89 Chapter 4 On the opiimizalion problems of some classes of sc perfect graphs Proof. By Theorem-4.6, it is clear that if {x, y) is an even pair in G then replacing x and ^^ by a new vertex z and obtaining reduced graph has same size of maximum clique as G. Also every two-pair is an even pair, so the above result is also true when we merge two non adjacent vertices which are two-pair into a new vertex. Hence the Theorem, D We demonstrate algorithm-4.1 by giving an example of sc weakly chordal graph on 8 vertices as shown in rigure-4.1 V4 Let us input G in algonthm-4.1. Step-1 of algorithm-4.1 finds {vi, V2} as one of the two-pair in G. Step-2 computes //(v/)uN(v;) = {V4, V6, V7, v^} u {V5, V6, V7, H) ^ {^4, V5, V6,V7, v;;} = W. Step-3 rcplaccs vertex V| and V2 by a new vertex z^. Step-4 adds edges such as Z1V4, Z1V5, ziVd, Z|V7, Z|V8. Step-5 repeats the process untill their does not remain any non adjacent pair of vertices. Continuing this way the algorithm-4.1 finds the last graph in the sequence as of order 3. Therefore the maximum clique and minimum coloring of G is (o{G) = x{G) = 3. 90 Chapter 4 On ihe opiimizaiion problems of some classes of sc perfect praphs The overall procedure of finding maximunn clique and minimum coloring of G is shown in below figure-4.2. (V6V7 • Z2) <: (Z5.4 ZiVg) (Zj.<—ZiVj) (0 Figurc-4.2 Let us now consider the other two optimization problems i.e. maximum stable set and minimum clique cover for sc weakly chordal graphs. Actually, Hayward et al. [67] presented only one algorithm, which solves the maximum clique and minimum coloring problems. For the solution of other two problems i.e. maximum stable set and minimum clique cover, they only mentioned that 91 Chapter 4 On the oplimizalion problems of some classes of sc perfect graphs the same algorithm can be ain on the complement to get solution. No more discussion was there on these two problems, so to understand it in more detail we discuss here and propose an algorithm for the solution of maximum stable set problem for sc weakly chordal graphs, while giving a different approach as compare to [67]. So, to obtain this first we give the following Corollary, which is immediate from the Theorem-2.3 and remembering the fact that sc weakly chordal graphs have exactly n{n-l)/4 edges. Corollarv 4.8. Let G be a sc weakly chordal graph. Then each induced subgraph of G contains a two-pair. The following Lemma also ensures us for the existence of the complement of two-pair i.e. co-pair in sc weakly chordal graph. Lemma 4.9. Let G be a sc weakly chordal graph. Then each induced subgraph of G contains a co-pair. Proof. Since a co-pair is two-pair of the complement of a graph and we are considering sc weakly chordal graph, which is isomorphic to its complement. Hence the result, D The above result shows the importance of a co-pair in a sc weakly chordal graph, so it is natural to try to formulate the analogue of the Edge addition Theorem-2.15 using co-pair, which we state in Theorem-4.15. Before we state and prove Theorem-4.15, we present some Lemmas that will be useful in the proof of Theorem-4.15. Moreover in order to prove the following Lemma's we use the process of recognition of an edge, which is co- 92 Chapter 4 On ihe opiimizalion problems of some classes of sc perfect graphs pair as follows: "Two adjacent vertices form a co-pair if and only if removing their common non-neighbors leaves the vertices in different components of the complement"[68]. Lemma 4.10. In a given graph G, an edge that belongs to a hole cannot be a co-pair. Proof. Suppose that G has a hole v\Vi...v\^, ^ > 5 as in figure-4.3(a) Let edge v\Vy be a co-pair. Neighbors of V| and v^ are V2 and v^.i respectively. The non- neighbors of vi are all the vertices except V2 and v\^ and non-neighbors of v^ are all the vertices except Vj and v^.], the common non-neighbors are V3 and V4. Now removing the common non-neighbors V3 and V4 from G and then considering the complement of the resultant graph which is shown in figure- 4.3(b). V2 >'k-i# V Vj • Vj (a) (b) Figurc-4.3 There is a path between vi and vi<, which passes through neighbors i.e. v^.i and vj. From figure-4.3(b) it is clear that V| and vi^ lie in the same connected component in the complement, which contradicts the assumption that V|Vk is a co-pair edge. This argument holds for any edge of hole. Thus no edge of a hole can be a co-pair. Hence the result, D 93 Chapter 4 On ihe optimization problems of some classes of sc perfect graphs The following Lemma shows the behavior of co-pair in antihole. In fact the proof is exactly on similar lines, but for completeness we include the proof Lemma 4.11. In a given graph G, an edge that belongs to an antihole cannot be a co-pair edge. Proof. Let us suppose that G has an antihole which is the complement of a hole on v\V2...v^, k> 5 (for ^ = 5 antihole is isomorphic to hole) as shown in figure-4.4. Assume that edge V|V4 is co-pair edge, now the neighbors of vi are all vertices except V2 and v^ and the neighbors of V4 are all vertices except V3 and v^;.|. The non-neighbors of vi and V4 are V2, vi^ and V3, vn respectively. Clearly there is no common non-neighbor of v, and v^, which implies that their is always a path between vi and V4 in the complement (one of them shown by doted lines in figure-4.4). Therefore vj and V4 lie in the same connected component in the complement, which contradicts that the edge V1V4 is co-pair. Thus V1V4 is not co-pair edge. The similar argument holds for every other edge of antihole. o Vk ^ ^2 Figurc-4.4 94 Chapter 4 On the optimization problems of some classes of sc perfect graphs Lemma 4.12. Let G be a cycle of length 5 and having only one chord. Then this chord cannot be a co-pair. Proof. Let G be a cycle of length 5 with edges viV2, V2V3, V3V4, V4V5 and V5V1. Let us suppose its only chord be V1V3, which is a co-pair. The non-neighbors of vertex vi and V3 are V4 and vj respectively. There is no common non-neighbors between v, and V3 so vertex vi and V3 always lie in the same connected component in the complement as shown by figure-4.5 (b). (a) (b) Figurc-4.5 This contradicts that V1V3 is a co-pair. Similarly other chords i.e. V1V4, V2V4, V2V5, V3V5 can also be shown that, they are not co-pairs. Hence the result, D Note. Since an antihole of length 5 is isomorphic to itself, so along the same line of the proof of above Lemma we can prove that, when a graph, which contains a Ck {k = 5) with exactly one chord. Then this chord will never be a co-pair. Lemma 4.13. Let G be a graph, which contains a cycle with edges V|V2, v2V},...,vy.]V\„ Vi;V| for k> 5. Let there be only one chord in this cycle. Then this chord cannot be a co-pair. Proof. Suppose that G has a cycle as shown in figure-4,6(a). 95 Chapter 4 On the opiimizaiion problems of some classes of sc perfect graphs V2 Vk V2 Vk-l Vk-l • Vk-l 4\1/ ^ Vk-l (a) (a) (b) Figure-4.6 Let us suppose that one of the chord V|V3 is a co-pair. The non-neighbors of vi and vj are V4, \\.\ and v^.,, vi^ respectively. So their common non-neighbor is v^.i. After removing this vertex vi^.i from G and considering the complement of the resultant graph the vertices V| and V3 lie in the same connected component, which contradicts the assumption that V1V3 is a co-pair. This argument holds for every other chord of the same type in the cycle. Let one of the other type of chord v,V4 be a co-pair. The non-neighbors of V| and V4 are v^.v^.i and V2, v^ respectively. Cleary there is no common non- neighbor between them, which implies that in the complement vertices V| and V4 always lie in the same connected component. This contradicts that V|V4 is a co-pair. By a similar argument it can be shown that every other chord of same type cannot be a co-pair. Therefore in either case the added chord is not a co- pair. Hence the result. o Lemma 4.14. Let G be any graph, which contains Ck with vertices V|V2...V|<, k > 6. Suppose there is exactly one chord between any two non-adjacent pair of vertices of Ck. Then this chord cannot be a co-pair. 96 Chapler4 On ihe opiimization problems of some classes of sc perfect eraphs Proof. Let G contains a C*, A: > 6. we have to show that the chord which is added is not a co-pair. Suppose it is added between vertices V2 and V3, as shown in figure-4.7(a). It is clear from figure-4.7(a) that each vertex of G except vj and V3 have non-neighbors at most 2, while vertex V2 and V3 have non-neighbors one each. ' (b) V, V; Vj V< (c) (a) Figure-4.7 Now as we know that the complement of CA is cycle of regular degree 2 (as shown in figure-4.7(b)), after deleting a chord from Ck, its complement becomes chordless path Py (as shown in rigure-4.7(c)). Clearly in any chordless path Pk {k> 3) no two end vertices have common neighbors, which implies that the vertices vj and V3 in G have no common non-neighbors. Therefore vi and V3 always lie in same connected component in the complement, which contradicts that V1V3 is a co-pair. This argument holds for any chord (exactly one in G) for any Ck {k > 6). Hence the result D In the above results we consider, hole and antihole with exactly one chord because this is the only case, when it is possible that after deleting a chord, a hole or antihole may be generated. On the other hand if hole and 97 Chapter 4 On the opiimizaiion problems of some classes of sc perfect eraphs antihole have more than one chord then deletion of a chord at a time certainly does not create a hole or antihole. We are now in a position to state and prove Theorem-4.15. Theorem 4.15. Let e^y be a co-pair of a graph G. Let Gi be the graph obtained from G by deleting e^y. Then G is weakly chordal if and only if G] is weakly chordal. Proof. For the necessary condition we have to show that the deletion of e^y does not destroy the property of weakly chordal graph. Deletion of Cxy from G will make G\ not weakly chordal only when Gi becomes either a hole or antihole. But from Lemma 4.12 and Lemma 4.13, we know that if deleted edge makes any graph hole then it will not be a co-pair edge i.e. we are deleting co- pair edge which cannot be a chord of a hole or antihole. Also from Lemma 4.12 and Lemma 4.14, it follows that the deleted edge which makes any graph antihole will not be a co-pair. Therefore deletion of e^y from G will always produce G\, which is weakly chordal. Now to prove the sufficient condition let G be not a weakly chordal, then either G is hole or antihole or contains them. Suppose G is a hole then from Lemma 4.10, no G\ will ever be produced. Similarly if G is antihole then from Lemma 4.11, again no G| will be produced. So if G| is weakly chordal then G is also. Hence the theorem. D Using Theorem-4.15, we suggest the following procedure for finding maximum stable set for sc weakly chordal graph via edge deletion method. The process is as follows: find a co-pair edge e,y and replace the two adjacent 98 CliaplL'r 4 On ihc opiimizalion problems of some classes of sc perfect craphs vertices x and j^' by a new vertex w with neighborhood N{w) = N(x)r\N(y) and then repeat the process. However an equivalent way of viewing the operation is as follows: mark j to w in the new graph, delete every edge incident on j, mark y as deleted and delete every edgexw such that UGN{X)- N{y). The algorithm is as follows. Algorithm-4.2: An algorithm for finding largest sable set and minimum clique cover for sc weakly chordal graph. Input: A sc weakly chordal graph G. Output: a{G) and6'(G) for sc weakly chordal graph G. Step-1: Find a co-pair Cxy of (7. and let V= set of vertices of G. Step-2; Compute N{\) -N(y) = U (let) Step-3: Replace vertex x with w and delete edges xu such that ueU. Step-4: Mark vertex y as deleted Step-5: Update V by replacing x andy by w. Step-6: Repeat the whole process until no co-pair found. Step-7: Return \V\^a{G)^e{G) End. Complexity. An edge is a co-pair and it can be tested in 0{n+m) time in step- I. Step-2 to step-5 can be done in linear time. Step-6 repeats the whole process n time. Therefore the overall time complexity of algorithm-4.2 is 0{{n+m)n) time. 99 Chaplcr4 On ihe opiimizalion problems of some classes of sc perfect f;raphs The correctness of algorithm-4.2 follows from the fact a{G) ^ 0){G) and from Theorem-4.6 and Theorem-4.7. Theorem 4.16. Algorithm-4.2 finds maximum stable set and minimum clique cover. We illustrate algorithm-4.2 by giving an example of sc weakly chordal on 9 vertices as shown in figure-4.8. Let us input sc weakly chordal graph G as shown in figure-4.8 in algorithm-4.2. Step-1 starts with vertex set V = {v\, vi, V3, V4, V5 v^, V7, v^, V9} and finds edge (vi, vs) is as one of the co-pair edge of G. Step-2 computes N{v\) - N{v^) = {v^, vj, va} - {vi, vi, V3, V4, vg) = {V7, Vii} = U. Step-3 replaces vertex vi with W) and deletes edges viVj and viVg. Step-4 deletes vertex vg and updates the vertex set V as (wi, V2, V3, V4, V5, V6, V7, V9}. Step-6 repeats the process, until no co-pair edge remains in G. Continuing in this way the algorithm-4.2 finally produces the maximum stable set of G as |K| = a(G)= 3, where V = {w], w^, W(,}. The overall procedure can be seen in figure-4.9. 100 Chaplcr 4 On ihc oplimi:anon problems of aome classes of sc perfect graphs (VlVg ^ W,) '^- {>'l, »'2. >'J, 1'4, I's, Vj, V'7, Vg, V,! V= \Wu V2, Vj, V4, V5, V6, V7, V,} V6 V3 J , ."^ .•V5- % (C) (d) (c) ^'= {^2, "',1, Vj, H-j, V,,} V= \wi, M'j, I'l, Vfc, V7, I',} ^'= {H":, V2, VJ, V5, vj, V7, V,} (VftW., •Ws) W6 W5 « • ^2 W5 « • W2 (V3W4 •Wj) (0 (g) K= {H'2, V,, Wj, (Vs) K= I H'2, M-s, M-^} Figurc-4.9 Cliapier 4 Un the optimization problems of some classes of sc perfect praphs 4,4 Optimization algorithms for the class of sc perfectly orderable graphs As a first step to understand this section, let us define, one of the natural way to color a graph i.e. greedy coloring algorithm [82], which is as follows: The first step is to impose an order < on the vertices Vj of a graph and then to progress through the vertices in the assigned order. If Vj < Vj then Vj appears before vj in the assigned ordering. At each vertex Vj we look at all the neighbors Vj of Vj such that Vj < Vj and assign V| the lowest possible color not assigned to one of these neighbors. This method is called greedy coloring algorithm. It is important to realize that the greedy algorithm does not guarantee that the coloring is an optimal coloring. So, the next logical question is: can we find a class of graphs for which the greedy coloring algorithm always produces an optimal coloring? In order to answer this question, Chvatal [26] proposed the concept of perfect order (<) and perfectly orderable graphs and gave the following result. Theorem 4.17 [261. Given a perfect order of a graph, the greedy coloring algorithm using the order computes a minimal coloring. Where an order (<) is perfect if for each induced subgraph, the chordless path on four vertices i.e. P4 with vertices a, h, c, d and edges ab, be, cd such {hdXa However, it is much difficult to find a perfect order of a graph. In [94] Middendorf and Pfeiffer proved that to decide whether a graph is perfectly 102 Chapter 4 On the optimizalion problems of some classes of sc perfect graphs orderable or have a perfect order is NP-complete. This has motivated researchers to study subclasses of perfectly orderable graphs in the hope that such an effort will lead to a better understanding of the combinatorial structure of the perfectly orderable graphs. For this purpose, we study the optimization problems for sc perfectly orderable graph classes such as sc quasi-chordal graphs and sc brittle graphs and propose a polynomial time algorithm, which solves the problem of minimum coloring in sc quasi-chordal and sc brittle graphs. 4.4.1 Optimization algorithm for sc quasi-chordal graphs and sc brittle graphs In order to solve the optimization problems for sc quasi-chordal graphs and sc brittle graphs, we need to find first their perfect order. Often, elimination order of any subclass of perfectly orderable graphs becomes perfect order for that graph. For example, simplicial vertex elimination order of chordal graph is known as simplicial order and is also perfect order. Unfortunately, it is neither true for sc quasi-chordal graphs nor i^or sc brittle graphs. For example consider the following sc quasi-chordal graph G which is also sc brittle as shown in figure-4.10. One of its elimination order is (V3 < V4 < V7 < vg < vi < V5 < V2 < v^). Now consider one of the induced Pa, of G which is [V3, v^, V2, V5] in which V3< V(„ vi < V5, this clearly implies that this is an obstruction (an obstruction is an ordered graph is the P^ [a, b, c, d] with a < b and d < c [108]) in G. Therefore this elimination order cannot be perfect order. 103 Chapter 4 Oil ihc opiimizauon problems o/ some classes of sc perfect graphs V7 V| v; y \, \^v G \ Vft • •vg Figurc-4.i0 However, after some modification on elimination order a of either sc quasi-chordal graph or sc brittle graph we can obtain perfect order a' from a. For obtaining this we use following process. Let G be a sc quasi-chordal graph, then the eliminated vertex of G is either a co-simplicial vertex or simplicial vertex. Thus a i.e. elimination ordering of G will contain co-simplicial and simplicial vertices. Now o' is easily obtained from o by placing first all co-simplicial vertices followed by all simplicial vertices. Similarly in the case of sc brittle graphs, the eliminated vertex is either no-mid or no-end, we also know that every simplicial vertex is no-mid and every co-simplicial vertex is no-end. So we can obtain a' from a in sc brittle graphs by first placing all no-mid vertices and then placing all no- end vertices. We are now in a position to discuss the following algorithm-4.3, which computes minimum coloring of sc quasi-chordal graphs and sc brittle graphs. The algorithm works as follows; after inputting the graph G for which we want to compute minimum coloring. Step-1, finds its elimination order, which can be 104 Chapter 4 On the opiimizalion problems of some classes of sc perfect eraphs done by either algorithm-3.2 (for sc brittle graphs) or algorithm-3.4 (for sc quasi-chordal graphs). In step-2 it obtains a' from a by doing necessary calculation. Now apply greedy coloring algorithm on a' in step-3. Step-4 returns the output ^'(G) of the input graph G. The algorithm is as follows. Algorithm-4.3: An Algorithm for finding minimum coloring for sc quasi- chordal graph and sc brittle graph. Input: A graph G either sc quasi-chordal or sc brittle graph. Output: /(G) of G. Step-1: Find a of G. (using algorithm-3.2 or 3.4) Step-2: Obtain a' from a. Step-3: Apply greedy algorithm on a'. Step-4: Return number of minimum color i.e. xf(G). End. Complexity. Since any elimination order a for sc quasi-chordal graphs and sc brittle graphs can be obtained in 0{n'm) time. Now this cr can be converted into perfect order a' in 0(«^) time. Step-3 can be done in 0{n+m) time. Hence overall time complexity of algorithm-4.3 is 0{n^m) time Theorem 4.18. Algorithm-4.3 finds minimum coloring of G. Proof. Since we are applying greedy coloring algorithm oner', which is perfect order. From Theorem-4.17, we know that applying greedy coloring algorithm on any perfect order will always produce minimum number of color. Hence the Theorem, n 105 Chupler 4 0)1 ihe opiimizalion prohli.'ms at some classes of sc perfect f;raphs We illustrate algorithm-4.3 by using the same example as shown in figure-4.10. Let us input this graph G in algorithm-4.3. Step-1 finds one of its elimination order using algorithm-3.4 aso- = (vj < v^ < v^ < v^ < v, < Vj < Vj < v^,). Now step-2 converts this elimination order a into perfect order cr' as a' = (v^ < V|, < V3 < V4 < V, < I'j < v, < V(J. Since a' is perfect order so applying greedy coloring algorithm on this order will necessarily produce optimal coloring for G, this is done in step-3, assigning colors 1, 2, 3... as follows: vertex V7 is the first in a' so obviously it takes color 1. The next vertex in a' is v^, which is neighbor of V7 so it cannot take color 1 so give it color 2. Next vertex V3 gets color 1 because it is non-neighbor of V7. Similarly vertices V5, vi and V(, get color 1, 2 and 3 respectively, as shown in figure- 4.11(b). Since all the vertices are colored, thus step-4 provides /(C) = 3 i.e. minimum coloring of G. Also it can be seen from the input graph G that it contains a triangle, hence optimal color cannot be less than 3. V2 c=C> ^'2(2) V5(.) V6(.M M(l) (a) (b) Figurc-4.11 106 Cliaplcr 4 On the optimiialion problems of some classes of sc perfect graphs 4.4.2 Algorithm for finding the solution of optimization problems for sc quasi-chordal graphs and sc brittle graphs without given perfect order To design a polynomial algorithm, which solves the optimization problem for perfectly orderable graphs without given perfect order is an open problem. Till today, there is no polynomial time algorithm for perfectly orderable graphs, this again shows the importance of the study of subclasses of perfectly orderable graphs. In this subsection we show that how to get solution of optimization problem for sc quasi-chordal graphs and sc brittle graphs without given perfect order. Since sc quasi-chordal graphs and sc brittle graphs both are the subclasses of sc weakly chorda! graphs so any optimizing algorithm for sc weakly chordal graphs will also work on these subclasses. If we input these two subclasses in either algorithm-4.1 or algorithm-4.2, then it will produce their maximum clique, minimum coloring, maximum stable set and minimum clique cover respectively. The important fact is that both the algorithms do not need any perfect order of input graph in input. Hence we have polynomial algorithms for sc quasi-chordal graphs and sc brittle graphs, which can solve their optimization problems without given a perfect order. So in view of above discussion we have following result. Theorem 4.19. Let G be a sc quasi-chordal graphs or sc brittle graphs, then maximum clique, minimum coloring, maximum stable set and minimum clique cover of G can be solved in polynomial time without given a perfect order. 107 CHAPTER FIVE Chapter 5 On the spectrum ol some classes of sc perfect f;raphs Chapter-5 On the spectrum of some classes ofsc perfect sraphs 5.1 Introduction The spectrum of graph has been widely used in graph theory to characterize the properties of graph and extract information from its structure. In this chapter we study spectral properties of sc perfect graphs, sc chordal graphs and sc weakly chordal graphs. We study bounds on eigenvalues, characterization of graphs via spectrum, cospectrality of graphs and energetic properties of above mentioned graph classes. In section 5.2 we obtain a result for Interlacing Theorem when the graph is sc. In section 5.3, we prove that the class ofsc and sc weakly chordal graphs cannot be determined through its given spectrum. In section 5.4, we prove that the least positive integer for which there exist non-isomorphic cospectral sc weakly chordal graph is 12. In section 5.5, we show that there exist non- isomorphic equi-energetic sc weakly chordal graphs with 8 vertices and there does not exist any non-isomorphic non-cospectral equi-energetic sc chordal graphs up to 13 vertices. In section 5.6, we obtain the upper bound for the energy of sc graphs and later prove that the maximal energy of sc graphs is always smaller than the maximal energy of general graphs on n vertices. In section 5.7, we first prove that no sc graph is hyper-energetic on /? < 8 vertices. 108 Cliaplcr 5 On the specirum of some classes of sc perfect graphs Moreover we also show that there does not exist any hyper-energetic sc chordal graphs up to 13 vertices and there exist hyper-energetic sc weakly chordal graphs on 12 vertices. 5.2 Sc perfect graph and Interlacing Theorem Eigenvalues are crucial for understanding the properties of graphs. An early problem in spectral graph theory was bounding the eigenvalues of a graph from below. One of the basic tools for bounding eigenvalues comes from matrix theory and is called Interlacing Theorem. In this section we obtain bounds on eigenvalues of sc graphs similar to Interlacing Theorem which is as follows. Theorem 5.1 (Interlacing Theorem) [4|. Let A be a real symmetric matrix with eigenvalues/l, > ^ ^•••^^7' having a principal submatrix with eigen values //,>//, ^... > //„.,. Then i, > //, > A, > //, >... > /l„_, > /y„_, > i„. The following result is well known in the field of matrix theory. Theorem 5.2 (Courant-Wevn 1361. Let HX) > XjiX) > ... > >n,W be the eigenvalues of a real symmetric matrix X. X'i A and B are real symmetric matrices of order n and C = .4 + B, then A„^.„(C) K-,-j{C)>KM)^X„_^{B) (5.2) where 0 < /, /', i+j+1 < n. In [35] D.Cvetkovic proved the following inequalities. 109 Chaplcr 5 On the spectrum of some classes of sc perfect graphs Theorem 5.3 [351. For any graph G with eigenvalues/l^ >X2 >...>X„, the following inequalities hold: X,+J,>-\ + nS,,,^ (5.3) /l„.,„+^"-,.i <-l + /7^„„,,,,- (5.4) Where 2 < ;' + j < n + I and S^,^ is the kronecker delta. Using the above form of the Courant-Weyl inequalities and Theorem- 5.3, we get an interesting result relating the eigenvalues of a graph with those of its complement to obtain an interlacing property. Theorem 5.4. Let G be a graph with /I, > /I2 >... ^ /l„ its eigenvalues and Ii>l2>...>A„ be the eigenvalues of G. Then the numbers -l-I„,-l-I„-i,...,-l-l2 interlace the eigenvalues of G, i.e. il >-\-1n >-l-Irt-l >...>-\-^2 >X„. Proof. From equation 5.3 we know that X.+Xj>-\ + "^2,,+y Putting / = 2, y = /? -1, we get ^2 + X„-\ >-l + /7^2 2^„^, ^-' + "^2,«.i (<^2.».i =0 asn?^!) Xj+X„-\ >-l or -\-X2 From equation 5.4 we know that /l,,.,^, +X„-i.\ ^-l + "^„+i,,+y Putting / = ^- 1, y= 1, we get A,_„,,^,+I,-ui<-l + '7(5„,,„_i„ or /l2+I«<-l + 0 Chapter S On the s/ieclruni of some classes of sc perfect eraphs J„ <-l-/L2 Combining the above two inequalities, we get that /ill < — 1 — /I2 ^ AH-1 If we repeat the above procedure in general, i.e. substituting i = k + 1, j = n - k in equation-5.3 and / = /? - k, j = k in equation 5,4, for all A: (1 and combining inequalities for all k together, we get -1-I„ >/l, >-l-A„-i >...>-l-l2 >A„ (5.5) If we now put / =1,7= « in equatiQn-5.3 we get or/l, >-l-A„ (5.6) Thus combining equation-5.5 and 5.6, we get A, >-l-I„ > A, >-l-I» 1 >...>-I-l2 > I„. This completes the proof. D The above result of course can be rephrased for sc graphs in the following form. Corollarv 5.5. Let G be a sc graph and Ai,, > Aj >... > A„ be its eigenvalues. Then the numbers -1 -/l„,-l - A,,,,,...,-!-A, interlace the eigenvalues of G i.e.A, >-I-A„>A, >-l-A„^, >...>-1-A, > A„. Chapicr 5 On the spectrum of some classes of sc perfect eraphs 5.3 Non-DS sc and sc weakly chordal graphs A graph G is said to be determined by its spectrum (DS for short), if any graph having the same spectrum as G is necessarily isomorphic to G. One important topic in the theory of graph spectra is how to determine whether a graph is DS or not? It was commonly believed that every graph is DS until the first counter-example (a pair of cospectral but non-isomorphic trees) was found by Collatz and Sinogowitz [32] in 1957. Since then, various construction of cospectral graphs have been studied extensively and a lot of results are presented in literature [37]. The most famous result was given by Schwenk [123] in 1973, stating that almost all trees are not DS with respect to adjacency spectrum. However it was found that proving that graphs are DS is much more difficult than just showing that they are not DS. Up to now, only few graphs with very special structures are known to be DS, for example some distance regular graphs, some line graphs, path graphs P,„ circuit graphs Q and complete graphs K,, are some example of graphs that are DS. In [144] Edwin and Haemers posed the following problem: Problem. Whether a graph or a graph class can be determined by their spectrum or not? In this section we show with the help of examples that the class of sc graphs and the class of sc weakly chordal graphs is non-DS with respect to adjacency matrix. Chapter 5 On lite speclrum of some classes o/ sc perfect graphs Example 5.6. Consider the following graph G\ with 9 vertices and its complement G\ as shown in flgure-5.1(a) and figure-5.l(b), respectively. We also note that both G\ and G, have same characteristic polynomial as given by X - 18x^ - 20x* + 55x^ + 64x^ - 58x^ - 56x^ + 17x + 12. We first, show that graph G\ is not sc graph. This can be easily seen as in G\ there exist two vertices V4 and V3 of degree 2, while in the complement of G| there exists only one vertex vy of degree 2. Therefore G\ is not isomorphic to G\ • Hence Gi is not sc graph. (a) Figurc-5.I Now consider the following sc graph Gj on 9 vertices given in figure-5.2. Its characteristic polynomial is also x'' - 18x^ - 20x' + 55x^ + 64x' - 58x^ - 56x^ + 17x + 12. 13 Chapter S On the speclrum of some classes of sc perfect ffraphs sj9 h < V, Vj GX V6 • VR / Figurc-5.2 V.1 Since both G\ and G2 have same characteristic polynomial thus it is not possible to decide on the basis of spectrum, whether the given graph or graph class is self-complementary or not? Example 5.7. Consider the graph G2 again as shown in rigure-5.2. This graph is also weakly chordal as well as self-complementary. Therefore using the similar argument as above, it follows that it cannot be decided on the basis of the given spectrum that the given graph or graph class is sc weakly chordal or not. 5.4 On the cospcctrality of sc chordal and sc weakly chordal graphs Finding pair of non-isomorphic graphs with the same spectrum is one of the earliest and continuing problem in spectral graph theory. Sridharan and Balaji solved this problem for sc chordal graphs in [133], by showing that the least positive integer for which there exist a non-isomorphic cospectral sc chordal graphs is 12. Recently Merajuddin et al. [92] also solved the similar type of problem for sc comparability graphs. In this section we generalize their study to the class of sc weakly chordal graphs and prove that when a sc weakly Chapter S On the spectrum of some classes of sc perfect graphs chordal graph is not chordal, then to exist a non-isomorphic cospectral sc weakly chordal graph the least positive integer is 12. The following result shows that on 8 vertices there does not exist any non-isomorphic cospectral sc weakly chordal graph which is not chordal. Theorem 5.8. No two non-isomorphic sc weakly chordal (but not chordal) graphs with 8 vertices are cospectral. Proof. The characteristic polynomials associated with the sc weakly chordal graphs on 8 vertices as shown in Figure-2.6 (chapter-2) are given below. Graph Characteristic polynomial a. 1 - 4A- - 6x' + 20.v' + 11 x' - 20.v' - 1 4A-" + / b. I 6A' + 40A^ + 21./ - 1 6A^ - 1 4A' + x** c. -16x - 1 2A' + 32A-^ + 25x' - 16x^ - 14x' + x' d. -3- 12X-2X' + 36A-'+3!.V'- 12x'- Mx'-fx'* e. 9 - 12x- 30x^ + 28X^-^ 35x'- 12A'- HX' + X*' f - 8x - 8x' + 32x' + 37x' - 8x' - 14x'' + x' g- I Ix^ + 40x' + 33x'' - 8A' -1 4A' + A' We note that none of the above polynomials can be obtained from any one of the other polynomials by multiplying with a real number. Hence no pair of non- isomorphic sc weakly chordal graphs with 8 vertices have the same spectrum. Hence the result, n Theorem 5.9. No two non-isomorphic sc weakly chordal (but not chordal) graphs with 9 vertices are cospectral. Proof. The sc weakly chordal graphs as shown in figure-2.7 (chapter-2) have 15 Chaplcr 5 On the spectrum of some classes of sc perfect graphs Graph Characteristic Polynomial a. ~5x+ 16x' + 6x' -40/ - 1 Ix' + SZx" + 18x' -/ b. - 32x^ - 68x' - 25x^ + 28x'' + 18x' - x' c. 32A-'' + 1 2JC' -eOx" - 29x^ + 28^" + 18x' -x^ d. A + 21X + 56x' + 14x' - 72x'' - 5 Ix' + 20x'' + 18x' - x' e. -12 - 17x + 56x' + 58x-' - 64x' - 55x' + 20x^ + 18x' - x^ f. -16x + 32x' + 44x' - 64x'' - 6 Ix' + 16x'' + 18x"' - x^ g- 24x' + 8x-' - 72x' - 57x' + 16x'' + 18x' - x' as their respective characteristic polynomials. So by the same argument as in Theorem-5.8, the result follows, n Theorem 5.10. There exist non-isomorphic cospectral sc weakly chordal (but not chordal) graphs with 12 vertices. Proof. Consider the two non-isomorphic sc weakly chordal graphs with 12 vertices as given in figure-5.3. These two graphs are non-isomorphic since the graph in rigure-5.3(a), has a vertex vi of degree 3 whereas the graph in figure- 5.3(b) does not have a vertex of degree 3. By the recognition algorithm-2.4, these graphs are also weakly chordal but not chordal, as graphs in figure-5.3(a) and figure-5.3(b) contain chordless cycle {vi, V3, V|2, VIQ} and {vi, V6, V12, vio} respectively. Figurc-5.3 (b) 116 Chapler 5 On the specirum of some classes of sc perfect fsraphs The characteristic polynomials associated with these sc weakly chorda! graphs are given below. Graph Characteristic Polynomial a. - 20A----I60x^ -- 391A:' - 272A-^ + 27!x'' + 484A' + 172x*' - 52A-'' - 33x'"+ x" b. - IQx" --i60A-^ -- 391.r'' - 272A-' + 27iA" + 484A^ + 172x*' - 52A' - 33x'"+ x" Since both the graphs have same spectrum, Hence they are cospectral. D Corollary 5.11. The smallest positive integer for which there exist cospectral sc weakly chordal (but not chordal) graphs is 12. Proof. Follows from Theorem-5.10. a 5.5 Equi-energetic sc chordal and sc weakly chordal graphs In this section we show that there exist non-isomorphic non-cospectral equi-energetic sc weakly chordal graphs on 8 vertices, while there do not exist any non-isomorphic non-cospectral sc chordal graphs up to 13 vertices. To obtain this let us first recall the definition of equi-energetic graphs. "If for two graphs G\ and Gj, the equality £(^1) = E{Gi) is satisfied, then Gi and Gi are said to be equi-energetic". Theorem 5.12. There exist non-isomorphic non-cospectral equi-energetic sc weakly chordal graphs with 8 vertices. Proof. Consider the two sc weakly chordal graphs as shown in figure-2.6(b) and figure-2.6(g). These two graphs are non-isomorphic, as the graph in figure- 2.6(b) has a vertex of degree 2 while graph in figure-2.6(g) does not have any vertex of degree 2. By Theorem-5.10 their spectrums are different so they are non-cospectral. The energies of these graphs are 11.1231, so these graphs are 117 Chapter 5 On the speclrum of some classes of sc perfect f;rophs equi-energetic. Hence there exist non-isomorphic non-cospectral equi-energetic sc weakly chordal graphs with 8 vertices, a Corollary 5.13. The smallest positive integer for which there exist non- isomorphic non-cospectral equi-energetic sc weakly chordal graph is 8. Proof. The proof of the Theorem follows from Theorem-5.12. n Since the graph as shown in figure-2.6(g), contains a chordless cycle, thus it is not chordal. Therefore for the class of sc chordal graphs we have following result. Theorem 5.14. There do not exist any non-isomorphic non-cospectral equi- energetic sc chordal graphs with 8 vertices. Proof. Energies of the 3 non-isomorphic non-cospectral sc chordal graphs with 8 vertices as shown in figure-2.6(a), figure-2.6(b) and figure-2.6(c) are 11.4003, 11.1231 and 12.0000 respectively. Since all the three graphs have different energies, hence the result, D Theorem 5.15. There do not exist any non-isomorphic non-cospectral equi- energetic sc chordal graphs with 9 vertices. Proof. Energies of the 3 non-isomorphic non-cospectral sc chordal graphs with 9 vertices shown in rigure-2.7(a), figure-2.7(b), figure-2.7(c) are 13.0712, 12.5262 and 13.4031 respectively. Since all the three graphs have different energies, hence the result, n 18 Chapter 5 On the spectrum of some classes of sc perfect eraphs However, there exist non-isomorphic equi-energetic sc chordal graphs on 12 vertices namely the graphs c and h and graphs j and k as given in figure-5.4 but unfortunately they are cospectral, which of course is not of our interest. Theorem 5.16. There do not exist any non-isomorphic non-cospectral equi- energetic sc chordal graphs with 12 vertices. Proof. Energies of the 16 non-isomorphic non-cospectral sc chordal graphs with 12 vertices shown in figure-5.4(a to p) are 19.1128, 18.6082, 19.4851, 20.5006, 21.3775, 19.6500, 20.6317, 18.6082, 19.4851, 19.9427, 19.9427, 21.7296, 17.8930, 19.3093, 20.2925 and 21.1694, respectively. Pair of graphs in figure-5.4(c, h) and in figure-5.4(j, k) have same energies but they are cospectral and all the remaining graphs have different energies. Hence the result. D Theorem 5.17. There do not exist any non-isomorphic non-cospectral equi- energetic sc chordal graphs with 13 vertices. Proof. Energies of the 16 non-isomorphic non-cospectral sc chordal graphs with 13 vertices shown in figure-5.5(a-p) are 20.9229, 20.3838, 21.2607, 22.0073, 22.8841, 21.2657, 22.2473, 20.3838, 21.2607, 21.5584, 21.5584, 23.1389, 19.3023, 20.7186, 21.7018 and 22.5787, respectively. Pair of graphs in rigure-5.5(b, d) and in figure-5.5(j, k) have same energies but they are cospectral and all the remaining graphs have different energies. Hence the result. D 119 Chapter 5 On the speclrum of some classes of sc perfect eraphs Since there does not exist any non-isomorphic non-cospectral equi- energetic graph up to 13 vertices, so we have the following conjecture. Conjecture 5.18. For n = 4p or n = 4p+\ vertices, no non-isomorphic non- cospectral equi-energetic sc chordal graphs exist. 5.6 Maximal energy of sc perfect graphs In 1971, McClelland [91] obtained the first upper bound for E{G) as follows: E{G) < ^Inw Since then, numerous other bounds for E{G) were found ( see [85], [155] and [156] ). Here we just list some upper bounds for E{G) which were obtained recently. In [85] Koolen and Moultan improved the McClelland bound and showed that for a graph G with n vertices and in edges £(G)<~ + J07-I) 2 in n \ n J and characterized those graphs for which these bounds are best possible. Then they also showed that E{G) is maximized when m = and gave the 4 following inequality for general graphs, which only depends on number of vertices n. E{G)<-(\ + ^) (5.7) 120 Chapter 5 On the speclrum of some classes of sc perfect eraphs Later Koolen and Moulton [84] again studied the above bounds and gave following bounds for bipartite graphs. ^2m^ 2m E{G)<2 + Jin-2) 2m-2 \ n J E{G)<^[42^^T). Zhou [156] gave bounds in terms of degree sequence {d\, dj,..., d^ for E{G) as T" d £{C)<^^^+J(/7-l) 2/77-- For obtaining the upper bound for the energy of sc graph we need the following result which was given by Schwenk and Wilson [121] Theorem 5.19 [1211. Let C be a connected graph with 2, tX^ >...>X„ eigenvalues. Then ^ Where A(G) is maximum degree of G. The following Corollary immediately follows from above Theorem. Corollary 5.20. Let G be a sc graph with /i, > A, > ... > X„ eigenvalues. Then n-\ X> Theorem 5.21. Let G be a sc graph on n vertices, then the inequality £(G)<^(l + y^ 2 ^ 121 Cliaplcr 5 On the specinim of some classes of sc perfect f^raphs holds. Proof. Let A, > A, >... > i„ are the eigenvalues of sc graph G. Then by Corollai7-5.20 ^n-]^ A,> \ ^ J We also know that the energy of a graph G is E{G)^i\X\ or E{G)-\X\^i\X, .(5.8) we also have 2 "("-1) '•=1 2 or (5.9) ' = 2 2 Now applying Cauchy-Schwartz inequality to the vectors (1,1,...,1) and (|/l2|,...,|A„|), and using equation (5.8) and (5.9), we obtain inequality n{n-\) ^^:1 [E{G)-X,Y <{n-\) «(/7-l) 2 or E{G) n{n -\) Now the function F{x)-x+\{n-\) -X is decreasing on the , tn-l n(n-\) n-\ /?• . intervalj 122 Chapter 5 On the speclrum nl some classes of sc perfect graphs (n-\\ Hence F{1^)< F must hold as well. From this fact, and inequality (5.9) 2 ; it immediately follows that /?(/?-1) (n-\ £(C)<^ + J(«-1) or £(G)<—(l + V^) (5.11). D We can see that for any positive integer n, the energy given by the equation-5.11 is always less than the energy given by the equation-5.7, so we have the following result. Theorem 5.22. The maximal energy of sc graph is always smaller than the maximal energy of general graph on same number of vertices. The above Theorem also provides a partial answer to problem raised by Richard Brualdi in [17]. "Which graph class has maximum energy on n vertices?" Corollary 5.23. On n vertices, no sc graph attains the maximum energy. For the maximal energy of sc chorda! and weakly chordal graph classes we have the following observation, shown by the graph in figure-5.6, up to 13 vertices; It is a straightforward observation that for /? < 13, the maximal energy of sc chordal graph is always less than the energy of sc weakly chordal graph. So we have £•„„„- (sc chordal graph graphs) < E„„„ (sc weakly chordai graphs) 123 Chapter 5 On the spectrum of some classes of sc perfect graphs We further observe that the maximal energy of sc weakly chordal graph is always less than the maximal energy of sc graph, so combining all these observations, we have the following conjecture; Conjecture 5.24. Forany number of vertices «, Emax (sc chordal graphs) < E„ax (^c weakly chordal graphs) < E^^x (sc graphs). Maximal Energy Graphs 50 I i Energy of General graphs 45 \ ^ 40 i ^5-Energy of sc graphs 35 : ••- Hyper-energetic S'SO i at ! Energy ofsc weakly chordal 1^25^ graphs 120 *-Energy ofsc chordal graphs CO = 15 10 5 0 5 8 9 12 13 16 17 Vertices of sc graphs Figure-5.6 124 Chapter 5 On I he sped rum of some classes of sc perfect graphs 5.7 Hyper-energetic sc chorda! and sc weakly chordal graphs In this section, we show that there does not exist any hyper-energetic sc chordal graph up to 13 vertices. Moreover we show that there exist hyper- energetic sc weakly chorda! graphs with 12 vertices. Let us first recall the definition of hyper-energetic graphs. "If the energy of a graph G is greater than 2(« - 1), i.e., energy of complete graph, then graph is said to be Hyper-energetic". Theorem 5.25. A sc graph G with n vertices (« < 8) cannot be hyper-energetic. Proof. Let G be a sc graph with n vertices. Then by Theorem-5.21 We have E{G)< (l + V^j (5.12) If the graph is not hyper-energetic; then following condition must holds. E{G)<'-^—\^ + 4^\]< 2(/7-l), or -—(l + V^)<2(«-i) (/7-l)(l + V«7])<4(/7-l) (l + VA7 +1) < 4 or Vn + i < 3 or «<8 Hence no sc graph is hyper-energetic with /? < 8. D Now for /? > 8, we have following results for sc chordal and weakly chordal graphs. Theorem 5.26. No sc weakly chordal graphs with 9 vertices are hyper- energetic. 125 Chapter 5 On ihe spectrum of some classes of sc perfect eranhs Proof. Energies of the 7 non-isomoqjhic non-cospectral sc weakly chorda! graphs with 9 vertices as shown in rigure-2.7 are 11.4003, 11.1231, 12.0000, 12.0547, 12.6603, 11.8215 and 11.1231, respectively We note that none of the graphs have energy more than 2{n-\) = 16. Hence no non-isomorphic sc weakly chordal graphs with 9 vertices are hyper-energetic, D Theorem 5.27. There exist hyper-energetic sc weakly chordal graphs with 12 vertices. Proof. Consider the sc graph on 12 vertices as shown in the figure-5.7. It is weakly chordal as decided by the recognition algorithm-2.4. Energy of this graph is 22.19, which is greater than 2(/7-l) = 22, so the graph is hyper- energetic. Hence the result, D Figure-5.7 For the class of sc chordal graphs we have the following results. Theorem 5.28. No sc chordal graphs with 9 vertices are hyper-energetic. Proof. Energies of the 3 non-isomorphic sc chordal graphs with 9 vertices as 126 Chapter 5 On the speclrum of some classes of sc perfect f;raphs shown in figure-2.7(a), figure-2.7(b) and flgure-2.7(c) are 13.0712, 12.5262 and 13.4031, respectively. We note that none of the graphs have energy more than 2(«-l) = 16. Hence no non-isomorphic sc chordal graphs with 9 vertices are hyper-energetic, D Theorem 5.29. No sc chordal graphs with 12 vertices are hyper-energetic. Proof. The energies of the 16 non-isomorphic sc chordal graphs with 12 vertices as shown in rigure-5.4(a-p) are 19.1128, 18.6082, 19.4851, 20.5006, 21.3775, 19.6500, 20.6317, 18.6082, 19.4851, 19.9427, 19.9427, 21.7296, 17.8930, 19.3093, 20.2925 and 21.1694, respectively. We note that none of the graphs have energy more than 2(/;-l) = 22. Hence no non-isomorphic sc chordal graphs with 12 vertices are hyper-energetic, D Theorem 5.30. No sc chordal graphs with 13 vertices are hyper-energetic. Proof. The energies of the 16 non-isomorphic sc chordal graphs with 13 vertices as shown in figure-5.5(a-p) are 20.9229, 20.3838, 21.2607, 22.0073, 22.8841, 21.2657, 22.2473, 20.3838, 21.2607, 21.5584, 21.5584, 23.1389, 19.3023, 20.7186, 21.7018 and 22.5787, respectively. We note that none of the graphs have energy more than 2{n-\) = 24. Hence no non-isomorphic sc chordai graphs with 13 vertices are hyper-energetic, D Since there does not exist any non-isomorphic hyper-energetic sc chordal graph up to 13 vertices, so we have the following conjecture. Conjecture 5.31. For n = 4p or n = 4p+l vertices, No non-isomorphic sc chordal graphs are hyper-energetic. 127 Chapter S On I he spectrum of some classes of sc perfect graphs 16 non isomorphic sc chordal graphs on 12 vertices (c) (b) (d) (e) (i) (I') Figurc-5.4 (continue) 128 Chapter 5 On the speciriim of some classes of sc perfect graphs (I) (m) (n) (0) Chapter 5 On the spectrum ol' some classes of sc perfect eraphs 16 non isomorphic sc chordal graphs on 13 vertices (d) (c) (0 Fifjurc-5.5 (continue) 130 Chapter S On the spectrum of some classes of sc perfect eraphs (m) (n) (0) (P) Fi};urc-5.5 131 REFERENCES References References 1. R. Alter, A character-isation of self-complementary graphs of order 8, Portug. Math., 34(1975) 157-161. 2. 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